Stiefel-Whitney class of complex projective spaces

A topological complex vector bundle $E$ on a manifold $M$ has Chern classes $c_i(E)\in H^{2i}(M;\mathbb Z)$ and its underlying real vector bundle $E_\mathbb R$ has Stiefel-Whitney classes $w_j(E_\mathbb R)\in H^{j}(M;\mathbb Z/2)$.
The relation betwen both could not be more idyllic:

$\bullet $ The odd Stiefel-Whitney classes of $E_\mathbb R$ vanish: $w_{2j+1}(E_\mathbb R)=0$.
$\bullet \bullet$ The canonical map $H^{2i}(M;\mathbb Z)\to H^{2i}(M;\mathbb Z/2)$ sends $c_i(E)$ to $w_{2i}(E_\mathbb R)$.

In the case of $E=T_{\mathbb P^n(\mathbb C)}$ we have $c_i(T_{\mathbb P^n(\mathbb C)})=\binom{n+1}{i}\in H^{2i}(P^n(\mathbb C);\mathbb Z)=\mathbb Z$ for $i=0,1,\dots,n$ and $c_i(T_{\mathbb P^n(\mathbb C)})=0$ for $i\gt n$.
Thus reducing mod. $2$ we get $w_{2i}((T_{\mathbb P^n(\mathbb C)})_\mathbb R)=\binom{n+1}{i} \operatorname{mod. 2}\in H^{2i}(P^n(\mathbb C);\mathbb Z/2)=\mathbb Z/2$ for $i=0,1,\dots,n$ and all the other Stiefel-Whitney classes of $(T_{\mathbb P^n(\mathbb C)})_\mathbb R$ are zero.

Bibliography
Milnor-Stasheff, Theorem 14.10 page 169 and Problem 14-B page 171.