The metric of a $2$-torus can be written as \begin{align} ds^2&=\frac{e^\psi}{\tau_2}|dy_1-\tau\,dy_2|^2\\ &=\frac{e^\psi}{\tau_2}(dy_1^2-2\tau_1dy_1dy_2+|\tau|^2dy_2^2) \end{align} where $e^\psi$ parametrizes the the area of the torus and $\tau=\tau_1+i\tau_2$ is the complex structure of the torus. Can this be generalized to the metric of an $n$-torus (with $n\in\mathbb{N}$)?
Thanks in advance!