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For $n$ a power of two, we give a complete description of the cohomologyalgebra $H^*(\widetilde G_{n,3};\mathbb Z_2)$ of the Grassmann manifold$\widetilde G_{n,3}$ of oriented $3$-planes in $\mathbb R^n$. We do this byfinding a reduced Gr\"obner basis for an ideal closely related to thiscohomology algebra. Using this Gr\"obner basis we also present an additivebasis for $H^*(\widetilde G_{n,3};\mathbb Z_2)$.

Gr\"obner bases in the mod $2$ cohomology of oriented Grassmann manifolds $\widetilde G_{2^t,3}$