Suppose two integers $n< m$, and one is given a matrix subspace of $n\times m$ complex matrices, says $S$.

I am asking for algorithms or conditions which can answer the following question:

Whether every non-zero element of $S$ has rank $n$?

Notice that any $n\times m$ complex matrix has rank less or equal to $n$, here we want the rank of the space is exactly $n$.