  We obtain a combinatorial expression for the coefficients of the boundary map
of real isotropic and odd orthogonal Grassmannians providing a natural
generalization of the formulas already obtained for Lagrangian and maximal
isotropic Grassmannians. The results are given in terms of the classification
into four types of covering pairs among the Schubert cells when identified with
signed $k$-Grassmannian permutations. It turns out that these coefficients only
depend on the positions changed over each pair of permutations. As an
application, we give an orientability criterion, exhibit a symmetry of these
coefficients and, compute low-dimensional homology groups.
