Let $i:G_k(\mathbb{R}^n)\to G_k(\mathbb{R}^\infty)$ be inclusion of grassmannians.
Then
$H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$.
$ H^*(G_k(\mathbb{R}^n);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar w_{n-k+1},\bar w_{n-k+2},\cdots,\bar w_{n})$.
(c.f. An additive basis for the cohomology of real Grassmannians, Lecture Notes in Mathematics Volume 1474, 1991, pp 231-234. )
Let $i^*: H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)\to H^*(G_k(\mathbb{R}^n);\mathbb{Z}_2)$ be the induced homomorphism in cohomology.
Is $i^*$ the quotient map $q: \mathbb{Z}_2[w_1,\cdots,w_k]\to \mathbb{Z}_2[w_1,\cdots,w_k]/(\bar w_{n-k+1},\bar w_{n-k+2},\cdots,\bar w_{n})$?