We derive governing equations that determine a full polarization state of transversely two-dimensional spatial solitons in a bulk anisotropic medium with the second-order nonlinearity. Based on nonlinear vectorial Maxwell's equations and approximation of slowly varying envelopes, our approach describes also lowest-order nonparaxial effects, however the most important factor governing radiation polarization is the medium anisotropy. This factor results in mixing of orthogonal components of electric field of quadratic soliton that consists of coupled beams at the fundamental frequency and its second harmonics. For the case of weak anisotropy we determine the soliton polarization state by a perturbation method; it turns out that it is elliptical and changing over the soliton transverse section. The approach allows generalization to the case of optical parametric oscillators.