I know $H^*(G_n(\mathbb{R}^{\infty});\mathbb{Z}_2)=\mathbb{Z}_2[w_1,...,w_n]$ where $G_n(\mathbb{R}^{\infty})$ is the Grassmannian manifold of all $n$-planes in $\mathbb{R}^{\infty}$, $w_i$ is the i-th Stiefel-Whitney class for the universal bundle $\gamma^n$.
I would appreciate it if anyone can tell me what is $H^*(\tilde{G_n}(\mathbb{R}^{\infty});\mathbb{Z}_2)$, where $\tilde{G_n}(\mathbb{R}^{\infty})$ is the oriented Grassmannian, and most preferably provide me a reference or a way to infer this from the knowledge of $H^*(G_n(\mathbb{R}^{\infty});\mathbb{Z}_2)$.
You can take a look at the book Characteristic Classes by John Milnor & James D. Stasheff (which you can find under the following link).
In Theorem 12.4 (on page 146) they use the Gysin sequence (introduced some pages earlier) to conclude that $$H^*(\tilde{G_n}(\mathbb{R}^\infty); \mathbb{Z}_2) \cong \mathbb{Z}_2[w_2(\tilde{\gamma^n}), \ldots, w_n(\tilde{\gamma^n})]$$ where $\tilde{\gamma^n}$ is the oriented universal bundle on the oriented Grassmannian $\tilde{G_n}(\mathbb{R}^\infty)$.