We present a method for expressing a potential energy surface (PES) for polyatomic molecules as an interpolation of local Taylor expansions in internal coordinates. This approach extends and replaces an earlier method which was only directly applicable to molecules of no more than four atoms. In general, the local Taylor expansions are derived from ab initio quantum calculations. Here, the methodology is evaluated by comparison with an analytic surface for the reactions H+CH(4)reversible arrow H-2+CH3. Approximately 1000-1300 data points are required for an accurate 12-dimensional surface which describes both forward and backward reactions, at the energy studied.