There is a theorem due to Sato which says that any vector bundle over $\mathbb CP^{\infty}$ decomposes as a direct sum of line bundles (which is a generalization of Grothendieck's result over $\mathbb CP^1$). I am wondering if there is a similar theorem for vector bundles over $\mathbb RP^{\infty}$ or a criterion for splitting. In my case I have a rank $2n$ grassmann bundle over $\mathbb RP^{\infty}$. Does it split as a direct sum of line bundles ? Thanks.



Vector bundles over $RP^{\infty}$
You are essentially asking about the set $[B\mathbb{Z}/2,BSO(2n)]$. This bijects with the set of conjugacy classes of homomorphisms from $\mathbb{Z}/2$ to $SO(2n)$, or in other words real, oriented representations of $\mathbb{Z}/2$. Essentially the same thing works for $[BP,BG]$ whenever $P$ is a finite $p$-group and $G$ is a compact connected Lie group. In fact, one can describe the full homotopy type of the space $\text{Map}(BP,BG)$: it is the classifying space of the groupoid of homomorphisms $P\to G$, and conjugacies between them. In this strong form the theorem is due to Dwyer and Zabrodsky. There are probably earlier references for the special case that you need. There is an exposition with references at <http://www.math.purdue.edu/~wilker/papers/bzpztobg.pdf>. 


I am not sure exactly what to say if you want $O(2n)$ instead of $SO(2n)$, but I expect that there is some kind of straightforward reduction.


