This is discussed in Stanley's paper *Some combinatorial aspects of the Schubert calculus*. Corollary 3.7 says that under the natural isomorphism given by the Borel presentation of $H^*(G/P)$ which sends an ordinary Schur function $s_{\lambda}$ to the class of the Schubert variety $X_{\lambda}$, a skew Schur function $s_{\lambda / \mu}$ is sent to the class of the Richardson variety $X_{\lambda}^{\mu}$.

Note that Stanley calls Richardson varieties *skew Schubert varieties* in this paper. Unfortunately, the version of the paper that is online at Stanley's website has some printing defects that make some of the pages illegible.