  Matrix Schubert varieties are the closures of the orbits of $B\times B$
acting on all $n\times n$ matrices, where $B$ is the group of invertible lower
triangular matrices. Extending work of Fulton, Knutson and Miller identified a
Gr\"obner basis for the prime ideals of these varieties. They also showed that
the corresponding initial ideals are Stanley-Reisner ideals of shellable
simplicial complexes, and derived a related primary decomposition in terms of
reduced pipe dreams. These results lead to a geometric proof of the
Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other
applications. We define skew-symmetric matrix Schubert varieties to be the
nonempty intersections of matrix Schubert varieties with the subspace of
skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe
a natural generating set for the prime ideals of these varieties. We then
compute a related Gr\"obner basis. Using these results, we identify a primary
decomposition for the corresponding initial ideals involving certain
fpf-involution pipe dreams. We show that these initial ideals are likewise the
Stanley-Reisner ideals of shellable simplicial complexes. As an application, we
give a geometric proof of an explicit generating function for symplectic
Grothendieck polynomials. Our methods differ from Knutson and Miller's and can
be used to give new proofs of some of their results, as we explain at the end
of this article.
