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

So we can apply Künneth Theorem right away:

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

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