We present rigorous, nonperturbative, purely numerical solutions of the Rayleigh equations for the scattering of p- and s-polarized light from a two-dimensional randomly rough perfectly conducting surface. The solutions are used to calculate the reflectivity of the surface, the mean differential reflection coefficients, and the full angular distribution of the intensity of the scattered field. These results are compared with the results of our earlier rigorous numerical solutions of the Stratton-Chu equations [Phys. Rev. A {\bf 82}, 013806(1-13) (2010)], and very good agreement is found.