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

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