Summary: We offer a possible physical explanation for the origin of the electron spin and the related antisymmetry of the wave function for a two-electron system, in the framework of nonrelativistic quantum mechanics as provided by linear stochastic electrodynamics. A consideration of the separate coupling of the electron to circularly polarized modes of the random electromagnetic vacuum field, allows to disclose the spin angular momentum and the associated magnetic moment with a \(g\)-factor 2, and to establish the connection with the usual operator formalism. The spin operator turns out to be the generator of internal rotations, in the corresponding coordinate representation. In a bipartite system, the distinction between exchange of particle coordinates (which include the internal rotation angle) and exchange of states becomes crucial. Following the analysis of the respective symmetry properties, the electrons are shown to couple in antiphase to the same vacuum field modes. This finding, encoded in the antisymmetry of the wave function, provides a physical rationale for the Pauli principle. The extension of our results to a multipartite system is briefly discussed.