For generic (not necessarily symmetric) $m\times n$ matrices over a set of $k$ elements, the number of those with pairwise distinct columns and rows is
$$\sum_{i=0}^m\sum_{j=0}^n s(m,i)\cdot s(n,j)\cdot k^{i\cdot j},$$
where $s(,)$ are Stirling numbers of first kind with sign.