Say I have an $n$-by-$n$ non-singular matrix $A$ all of whose diagonal entries are $0$. We call an $m$-by-$m$ minor of $A$ *good* if its set $I$ of row indices and its set $J$ of column indices ($I,J\subset \{1,2,\dotsc,n\}$) are disjoint. Can one give a good lower bound on the size $m$ of the largest non-singular good minor of $A$?

(Perhaps $m = \lfloor n/2\rfloor$?)

EDIT: All right, so obviously there aren't enough conditions - the answer is too easy. What if $A$ is antisymmetric?