Computing the Stiefel-Whitney classes of the tautological bundles $\gamma^n\rightarrow Gr(n,\mathbb R^\infty)$ I was reading about the Stiefel-Whitney classes construction using Grassmannians on Wikipedia's page.
It seems to me that all the axioms can be deduced once the $k$-th Stiefel-Whitney class of a rank $n$ vector bundle $E\rightarrow X$ is defined as $w_k(E)=f^*w_k(\gamma^n)$, where $f:X\rightarrow Gr(n,\mathbb R^\infty)$ has the homotopy class corresponding to $E$.
My question is, can the $w_k(\gamma^n)$ be computed using only the normalization axiom (and perhaps assuming that trivial bundles have no non-zero classes for $k\geq 1$)?
And if so, how can it be done?
In a different way, how is the Stiefel-Whitney class of $\gamma^n$ computed?
What is the idea?
Because Wikipedia says that $w_k(\gamma^n)$ corresponds to the generator $x_k\in H_k(Gr(n,\mathbb R^\infty),\mathbb Z/2\mathbb Z)$, but no explanation of why that is true is given.
 A: Yes, the normalization axiom suffices. The method to calculate higher Stiefel-Whitney classes from $w_1$ is called the splitting principle. The splitting principle says that for any space $X$ and vector bundle $E \rightarrow X$, there is a space $Z$ and map $Z \rightarrow X$ such that the map $H^*(X) \rightarrow H^*(Z)$ is injective, and the pullback of $E$ splits as a sum of line bundles.
With this fact, we can use the Cartan formula plus definition of $w_1$ to calculate the SW classes of the splitting of $E$ (which is a vector bundle over $Z$), and then we know this uniquely determines the SW classes of $E$ by naturality plus the injectivity of $H^*(X) \rightarrow H^*(Z)$.
The construction of the map $Z \rightarrow X$ is straightforward. Inductively, we define $X'$ to be the projectivization of $E \rightarrow X$, i.e. the points of $X'$ are the 1-dimensional subspaces of the fibers of $E \rightarrow X$. Then the pullback along the projection $X' \rightarrow X$ splits over a point $\ell_x \in X'$ as $\ell_x + F_x/\ell_x$. Now repeat this process with $X'' \rightarrow X'$.
