The canonical metric on the space of lattices in $\mathbb R^n$

Question

Let $\Lambda_1, \Lambda_2$ be two lattices on $\mathbb R^n$, (is there/)what is a canonical way to define the distance between $\Lambda_1, \Lambda_2$ such that the metric gives the topology on the space of lattices $\mathcal{L_n}$ on $\mathbb R^n$ that coincides with the quotient topology on $GL(n,\mathbb R)/GL(n,\mathbb Z)$ (please let me know if it should be another group instead of $GL(n,\mathbb R)/GL(n,\mathbb Z)$)?