For $Gl_n$, I like to use the group presentation found in Springer's Linear Algebraic Groups chapters 8-9. Basically the generators are digonal matrices and unipotent matrices (triangular with all 1's on diagonal). If you choose your $A_1, ..., A_n$ to be a nice subset of the generators, then often elements will have a unique expression as a product of those generators. Decomposition theorems (e.g. Bruhat) are also useful. So here you may want to restrict your $m$ to be less than $n$ say.
Am I understanding the question right - you have a fixed set of $n$ matrices and want the (infinitely many) possible products to be distinct?