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 $H^{\ast }({\tilde{G}}_{n,3};{\mathbb{Z}}_2)$ of the Grassmann manifold ${\tilde{G}}_{n,3}$ of oriented $3$-dimensional subspaces of ${\mathbb{R}}^n$. A complete description of $H^{\ast }({\tilde{G}}_{n,3};{\mathbb{Z}}_2)$ 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 ${\mathbb{Z}}_2$.