Can a square matrix be factored by non square matrices

Question

Given a square matrix $A\in M_{n\times n}(\mathbb{R})$. Can we always factor $A$ as a product of two non square matrices $B\in M_{n\times m}(\mathbb{R})$ and $C\in M_{m\times n}(\mathbb{R})$ such $A=BC$. And if its possible and we exclude multiples of $B,C$ is this decomposition unique. Thanks in advance