In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference sA - sB of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Our conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for sA = sB, and we deduce a strengthening of their result as a special case.