In this paper we consider the usual Einstein-Hilbert action of General Relativity (GR), but with a modified variational principle. To characterize a spacetime, we need to introduce two objects which are, in principle, independent, namely: the metric and the connection. In GR, the fundamental field is the metric tensor that completely determines the spacetime structure because the connection is nothing but the Levi-Civita connection generated by the metric. In the so-called Palatini formalism, the metric and the connection are treated as independent objects. However, for the case of the Einstein-Hilbert action, the equations of motion impose the connection to be the Levi-Civita connection of the metric. Thus, for such a simple action, both variational principles lead to the same physical theory. This is not the case for more general actions. In this paper, we consier an alternative variational principle. As in the Palatini formalism, we we treat the metric and the connection as independent objects. However, we further assume that the connection is generated by an additional rank 2 tensor, that we call potential metric. The way in which this potential metric generates the connection is by imposing that it is covariantly conserved. A key feature of this variational principle is that the potential metric is not assumed to be symmetric so that its antisymmetric parte will produce non-vanishin torsion. We obtain the linearized version of the theory, consisting of the usual GR action plus an additional graviton field and a 2-form field.