Two flat tori are isometric iff their lattices are isometric?

Question

Say $T_1,T_2$ are flat tori defined as quotients of $\mathbb{R}^n$ by the lattices $\Gamma_1,\Gamma_2$, resp. It is easy to prove that an isometry $\mathbb{R}^n\to\mathbb{R}^n$ sending $\Gamma_1\to\Gamma_2$ must descend to an isometry $T_1\to T_2$. How can one prove the converse: that if there exists an isometry $T_1\to T_2$, one can lift it to an isometry $\mathbb{R}^n\to\mathbb{R}^n$ (which sends $\Gamma_1\mapsto\Gamma_2$)?