Deteterminant of a zero-diagonal matrix with 1s and -1s

Question

Consider an $n$ by $n$ whose diagonal has all zero entries, everything over the diagonal is -1 and under the diagonal is 1. Is there a neat way to compute the determinant? Looks like it is either 0 or 1 depending whether $n$ is even or odd and could make it into an inductive proof but maybe there is a better way?