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The metrics are described by the following matrices:

$$ g_{\mathbb{R}}=\begin{pmatrix} \alpha_{11} & \beta_{11} & \cdots & \alpha_{1n} & \beta_{1n}\\ -\beta_{11} & \alpha_{11}& \cdots & -\beta_{1n} & \alpha_{1n}\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ \alpha_{n1} & \beta_{n1} & \cdots & \alpha_{nn} & \beta_{nn}\\ -\beta_{n1} & \alpha_{n1}& \cdots & -\beta_{nn} & \alpha_{nn}\\ \end{pmatrix} $$

$$ g_{\mathbb{C}}=\begin{pmatrix} 0 & g_{1\bar{1}} & \cdots & 0 & g_{1\bar{n}}\\ g_{\bar{1}1} & 0& \cdots & g_{\bar{1}n} & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots\\ 0 & g_{n\bar{1}} & \cdots & 0 & g_{n\bar{n}}\\ g_{\bar{n}1} & 0& \cdots & g_{\bar{n}n} & 0\\ \end{pmatrix} $$

$$ C=\begin{pmatrix} \dfrac{1}{2} & \dfrac{1}{2} &&&\\ \dfrac{-\mathrm{i}}{2} & \dfrac{\mathrm{i}}{2} &&&\\ &&\ddots&&\\ &&&\dfrac{1}{2} & \dfrac{1}{2}\\ &&&\dfrac{-\mathrm{i}}{2} & \dfrac{\mathrm{i}}{2}\\ \end{pmatrix} $$

$$ g=\begin{pmatrix} g_{1\overline{1}} & \cdots & g_{1\overline{n}} \\ \vdots&\ddots&\vdots\\ g_{n\overline{1}} & \cdots & g_{n\overline{n}}\\ \end{pmatrix} $$

$g_{\mathbb{R}}$ expresses the metric with respect to $\left\{\dfrac{\partial}{\partial x^1},\dfrac{\partial}{\partial y^1},...,\dfrac{\partial}{\partial x^n},\dfrac{\partial}{\partial y^n}\right\}$, whereas $g_{\mathbb{C}}$ expresses it with respect to $\left\{\dfrac{\partial}{\partial z^1},\dfrac{\partial}{\partial \bar{z}^1},...,\dfrac{\partial}{\partial z^n},\dfrac{\partial}{\partial \bar{z}^n}\right\}$. $C$ is the basis change matrix, and $g_{\mathbb{C}}=C^{T}g_{\mathbb{R}}C$, as may be checked blockwise:

$$ \begin{pmatrix} \dfrac{1}{2} & \dfrac{-\mathrm{i}}{2}\\ \dfrac{1}{2} & \dfrac{\mathrm{i}}{2}\\ \end{pmatrix} \begin{pmatrix} \alpha_{jk} & \beta_{jk} \\ -\beta_{jk} & \alpha_{jk} \\ \end{pmatrix} \begin{pmatrix} \dfrac{1}{2} & \dfrac{1}{2}\\ \dfrac{-\mathrm{i}}{2} & \dfrac{\mathrm{i}}{2}\\ \end{pmatrix}= \begin{pmatrix} \dfrac{1}{2} & \dfrac{-\mathrm{i}}{2}\\ \dfrac{1}{2} & \dfrac{\mathrm{i}}{2}\\ \end{pmatrix} \begin{pmatrix} \dfrac{\overline{h_{jk}}}{2} & \dfrac{h_{jk}}{2}\\ \dfrac{-\mathrm{i}\bar{h}_{jk}}{2} & \dfrac{\mathrm{i} h_{jk}}{2}\\ \end{pmatrix}= \begin{pmatrix} 0 & \dfrac{h_{jk}}{2}\\ \dfrac{\overline{h_{jk}}}{2} & 0\\ \end{pmatrix}= \begin{pmatrix} 0 & g_{j\bar{k}}\\ g_{\bar{j}k} & 0\\ \end{pmatrix} $$

In particular,

$$\det g_{\mathbb{C}}=\det g_{\mathbb{R}}(\det C)^2=\left(\dfrac{\mathrm{i}}{2}\right)^{2n}\det g_{\mathbb{R}}$$

Moreover, according to the properties of the determinant,

$$\det g_{\mathbb{C}}=(-1)^n\det\begin{pmatrix} g_{1\overline{1}} & \cdots & g_{1\overline{n}} \\ \vdots&\ddots&\vdots\\ g_{n\overline{1}} & \cdots & g_{n\overline{n}}\\ \end{pmatrix} \det\begin{pmatrix} g_{\overline{1}1} & \cdots & g_{\overline{1}n} \\ \vdots&\ddots&\vdots\\ g_{\overline{n}1} & \cdots & g_{\overline{n}n}\\ \end{pmatrix}$$

but since the last two matrices are transpose of each other,

$$\det g_{\mathbb{C}}=(-1)^n(\det g)^2$$

The volume form is therefore

$$ \begin{array}{rcl} \mathrm{d} V & = & \sqrt{\det g_{\mathbb{R}}}\dfrac{\partial}{\partial x^1}\wedge\dfrac{\partial}{\partial y^1}\wedge...\wedge\dfrac{\partial}{\partial x^n}\wedge\dfrac{\partial}{\partial y^n}\\ & = & \left(\dfrac{2}{\mathrm{i}}\right)^n\sqrt{\det g_{\mathbb{C}}}\left(\dfrac{\mathrm{i}}{2}\right)^n\dfrac{\partial}{\partial z^1}\wedge\dfrac{\partial}{\partial \bar{z}^1}\wedge...\wedge\dfrac{\partial}{\partial z^n}\wedge\dfrac{\partial}{\partial \bar{z}^n}\\ & = & \sqrt{\det g_{\mathbb{C}}}\dfrac{\partial}{\partial z^1}\wedge\dfrac{\partial}{\partial \bar{z}^1}\wedge...\wedge\dfrac{\partial}{\partial z^n}\wedge\dfrac{\partial}{\partial \bar{z}^n}\\ & = & \mathrm{i}^n\det g \dfrac{\partial}{\partial z^1}\wedge\dfrac{\partial}{\partial \bar{z}^1}\wedge...\wedge\dfrac{\partial}{\partial z^n}\wedge\dfrac{\partial}{\partial \bar{z}^n}\\ \end{array} $$

Also

$$ \begin{array}{rcl} \omega\wedge\overset{n}{\cdots}\wedge\omega\left(\dfrac{\partial}{\partial z^1},\dfrac{\partial}{\partial \bar{z}^1},...,\dfrac{\partial}{\partial z^n},\dfrac{\partial}{\partial \bar{z}^n}\right) & = & \mathrm{i}^n\left(\sum_{j_1\bar{k}_1}g_{j_1\bar{k}_1}\mathrm{d} z^{j_1}\wedge\mathrm{d} \bar{z}^{k_1}\right)\wedge\cdots\wedge\left(\sum_{j_n\bar{k}_n}g_{j_n\bar{k}_n}\mathrm{d} z^{j_n}\wedge\mathrm{d}\bar{z}^{k_n}\right)\left(\dfrac{\partial}{\partial z^1},\dfrac{\partial}{\partial \bar{z}^1},...,\dfrac{\partial}{\partial z^n},\dfrac{\partial}{\partial \bar{z}^n}\right)\\ & = & \mathrm{i}^n\sum g_{j_1\bar{k}_1}\cdots g_{j_n\bar{k}_n}\mathrm{d} z^{j_1}\wedge\mathrm{d}\bar{z}^{k_1}\wedge\cdots \wedge\mathrm{d} z^{j_n}\wedge\mathrm{d}\bar{z}^{k_n}\left(\dfrac{\partial}{\partial z^1},\dfrac{\partial}{\partial \bar{z}^1},...,\dfrac{\partial}{\partial z^n},\dfrac{\partial}{\partial \bar{z}^n}\right)\\ & & \qquad\text{[the term is zero unless \(\{j_1,...,j_n\}\) and \(\{k_1,...,k_n\}\) are permutations of \(\{1,...,n\}\)]}\\ & = & \mathrm{i}^n\sum_{\sigma\tau}(-1)^{\sigma}(-1)^{\tau}g_{\sigma(1)\overline{\tau(1)}}\cdots g_{\sigma(n)\overline{\tau(n)}}\\ & & \qquad[\rho=\sigma^{-1}\tau]\\ & = & \mathrm{i}^n n!\sum_{\rho}(-1)^{\rho}g_{1\overline{\rho(1)}}\cdots g_{n\overline{\rho(n)}}\\ & = & \mathrm{i}^n n!\det g\\ \end{array} $$

and therefore

$$ \omega^n=n!\mathrm{d} V$$