We consider a natural form of a unified theory of gravity and electromagnetism which was somehow missed at the time of intense search for such a unification. The basic idea of this unification is to use the symmetric metric and a non-symmetric connection as independent variables, which generalizes the so-called Palatini formalism. For the first time this idea was applied to the construction of the action for a unified theory of gravity and electromagnetism without matter only in 1978, but this result did not receive wide recognition. In this paper we propose a natural way to include matter in the form of classical particles into the unified theory. The trace of connection in the appearing theory can be naturally identified with the electromagnetic potential, and the Einstein-Maxwell equations with classical matter are reproduced. We compare this approach with the known ideas of unification and briefly discuss the perspectives of a further development of this approach.