In this paper, a novel volume integral equation for solving the Electroencephalography forward problem is presented. Differently from other integral equation methods standardly used for the same purpose, the new formulation can handle inhomogeneous and fully anisotropic realistic head models. The new equation is obtained by a suitable use of Green's identities together with an appropriate handling of all boundary conditions for the EEG problem. The new equation is discretized with a consistent choice of volume and boundary elements. Numerical results shows validity and convergence of the approach, together with its applicability to real case models obtained from MRI data.