The conductance of a contact, having a radius smaller than the Fermi wave length, on the surface of a thin metal film is investigated theoretically. It is shown that quantization of the electron energy spectrum in the film leads to a step-like dependence of differential conductance G(V) as a function of applied bias eV. The distance between neighboring steps in eV equals the energy level spacing due to size quantization. We demonstrate that a study of G(V) for both signs of the voltage maps the spectrum of energy levels above and below Fermi surface in scanning tunneling experiments.PACS: 74.55.+v Tunneling phenomena: single particle tunneling and STM; 85.30.Hi Surface barrier, boundary, and point-contact devices; 73.50.-h Electronic transport phenomena in thin films.