Electrophysiological modeling of Magneto- and Electro-encephalography (MEG and EEG) rely on accurate forward solvers that relate source activities to sensor measurements. In comparing a Boundary Element (BEM) and a Finite Element Method (FEM) for forward electroencephalography, in our early numerical experiments, we found the FEM to have a better accuracy than the BEM. This triggered a quest to improve the accuracy of Boundary Element Methods and led us to study the extended Green representation theorem. A fundamental result in potential theory shows that, up to an additive constant, a harmonic function is determined within a domain from its value on the boundary (Dirichlet condition), or the value of its normal derivative (Neumann condition). The Green Representation Theorem has been used in forward EEG and MEG modeling, in deriving the Geselowitz BEM formulation, and the Isolated Problem Approach. The extended Green Representation Theorem provides a representation for the directional derivatives of a piecewise-harmonic function. By introducing the normal current as an additional variable in the forward problem, we derive a new Boundary Element Method, which leads to a symmetric matrix structure: we hence call it the Symmetric BEM. Accuracy comparisons demonstrate the superiority of the Symmetric BEM to the FEM and to the classical BEM.