Cohomology rings of oriented Grassmann manifolds $\widetilde G_{2^t,4}$

We give a description of the mod 2 cohomology algebra of the oriented Grassmann manifold $\widetilde G_{2^t,4}$ as the quotient of a polynomial algebra by a certain ideal. In the process we find a Gr\"obner basis for that ideal, which we then use to exhibit an additive basis for $H^*(\widetilde G_{2^t,4};\mathbb Z_2)$.