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.