The metrics are described by the following matrices:

$g_{\mathbb{R}}$ 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:

In particular,

Moreover, according to the properties of the determinant,

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

The volume form is therefore

Also

and therefore