We define skew Schubert polynomials to be normal form (polynomial)
representatives of certain classes in the cohomology of a flag manifold.
We show that this definition extends a recent construction of Schubert
polynomials due to Bergeron and Sottile in terms of certain increasing
labeled chains in Bruhat order of the symmetric group.
These skew Schubert polynomials
expand in the basis of Schubert polynomials with nonnegative integer
coefficients that are precisely the structure constants of the cohomology of
the complex flag variety with respect to its basis of Schubert classes.
We rederive the construction of Bergeron and Sottile in a purely
combinatorial way, relating it to the construction of Schubert
polynomials in terms of rc-graphs.