Let $\xi=(E,p,B)$ be an $n$-dimensional vector bundle.
Let $f: B\to G_n(\mathbb{R}^\infty)$ be the classifying map.
Let $f^*: H^*(G_n(\mathbb{R}^\infty),\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_n]\to H^*(B,\mathbb{Z}_2)$ be the induced cohomology ring homomorphism.
What can we say about $f^*$?
Is it possible that $f^*=0$, i.e, maps everything to $0$?
Does $f^*$ have to be surjective?
$f^{\ast}$ is precisely the map that sends each universal Stiefel-Whitney class $w_1, w_2, \dots w_n$ to the actual Stiefel-Whitney classes of the bundle $E$. It must map the identity to the identity (so $f^0 : H^0 \to H^0$ can never be zero), but other than that, it can certainly map all of the $w_i$ to $0$; this is equivalent to the condition that the Stiefel-Whitney classes of $E$ vanish, and holds, for example, whenever $E$ is stably trivial. In particular there is no reason for $f^{\ast}$ to be surjective.