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The torus of dimension $n$ is defined as the product of $n$ circunferences:

$$\mathbb{T}^n= \mathbb{S} ^1\times \cdots \times \mathbb{S} ^1$$

So we can apply Künneth Theorem right away:

$$H^*(\mathbb{T}^n)=H^*(\mathbb{S} ^1)\otimes\cdots\otimes H^*(\mathbb{S} ^1)$$

Since $H^0(\mathbb{S} ^1_i)=\langle [1] \rangle$ and $H^1(\mathbb{S} ^1_i)=\langle [\mathrm{d}\theta_i] \rangle$, it turns out that

$$\{[\mathrm{d}\theta_{i_1}\wedge...\wedge\mathrm{d}\theta_{i_k}] :1\leqslant i_1 <...< i_k\leqslant n\}$$

is a basis of $H^k(\mathbb{T}^n)$. In particular,

$$H^k(\mathbb{T}^n)\simeq \mathbb{R} ^{\binom{n}{k}}$$