On the mod 2 cohomology algebra of oriented Grassmannians

For \(n\in \{2^t-3,2^t-2,2^t-1\}\) \((t\ge 3)\) we study the cohomology algebra \(H^*(\widetilde{G}_{n,3};{\mathbb {Z}}_2)\) of the Grassmann manifold \(\widetilde{G}_{n,3}\) of oriented 3-dimensional subspaces of \({\mathbb {R}}^n.\) A complete description of \(H^*(\widetilde{G}_{n,3};{\mathbb {Z}}_2)\) is given in the cases \(n=2^t-3\) and \(n=2^t-2,\) while in the case \(n=2^t-1\) we obtain a description complete up to a coefficient from \({\mathbb {Z}}_2.\)