Other possible answers:
1) You might want equations for the preimage of the Schubert variety in GL_n (i.e. the Schubert variety in Stiefel coordinates). These Their closures in M_n are matrix Schubert varieties, and their equations are given by Fulton in a paper in Duke Math J. in 1992 (not entirely sure about the year). Knutson and Miller say a lot more about these equations in their Annals of Math. paper (around 2003).
2) You might want equations for local affine neighborhoods in the Schubert variety. This is easy to get from (1) and the description of opposite Schubert cells as sets of matrices. Alex Yong and I work it out in a recent J. Algebra paper. One can derive a different (but somehow equivalent) set of local equations from standard monomial theory also.
