Imposing conditions on the ranks of upper-left corners of a skew-symmetric matrix defines a skew-symmetric matrix Schubert variety. We give Gröbner bases for the prime ideals of these varieties, and identify the corresponding initial ideals as the Stanley-Reisner ideals of certain explicit shellable simplicial complexes. These results are analogous to results of Knutson and Miller in the setting of ordinary matrix Schubert varieties, but the techniques are new, and can be used to give new proofs of some of their results. Based on joint work with Eric Marberg.