We propose a formal graph-theoretic model for studying the problem of matching rides online in a ride-sharing platform. Unlike most of the literature on online matching, our model, that we call {\em Online Windowed Non-Bipartite Matching} ($\mbox{OWNBM}$), pertains to online matching in {\em non-bipartite} graphs. We show that the edge-weighted and vertex-weighted versions of our model arise naturally in ride-sharing platforms. We provide a randomized $\frac{1}{4}$-competitive algorithm for the edge-weighted case using a beautiful result of Lehmann, Lehmann and Nisan (EC 2001) for combinatorial auctions. We also provide an $\frac{1}{2} (1 - \frac{1}{e})$-competitive algorithm for the vertex-weighted case (with some constraint relaxation) using insights from an elegant randomized primal-dual analysis technique of Devanur, Jain and Kleinberg (SODA 2013).