  We give a combinatorial expansion of the stable Grothendieck polynomials of
skew Young diagrams in terms of skew Schur functions, using a new row insertion
algorithm for set-valued semistandard tableaux of skew shape. This expansion
unifies some previous results: it generalizes a combinatorial formula obtained
in earlier joint work with L\'opez Mart\'in and Teixidor i Bigas concerning
Brill-Noether curves, and it generalizes a 2000 formula of Lenart and a recent
result of Reiner-Tenner-Yong to skew shapes. We also give an expansion in the
other direction: expressing skew Schur functions in terms of skew Grothendieck
polynomials.
