Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$

Question

I've come across mention of a bijection between lattices in $F^n$ ($F$ a field, in my case $\mathbb{C}(\!(t)\!)$) and elements of $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$, where $O$ is the ring of integers (in my case, $\mathbb{C}[[t]]$), but I haven't been able to find any references or complete exposition. I believe the bijection is given by sending $x \in \operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ to $x\Lambda$, where $\Lambda$ is the standard lattice in $O$ (so, I think, just $O$ itself). Does anyone know of a book or paper that might have more detail?