Line bundles over $\mathbb R P^2$

Question

As in this post, I'm continuing studying line bundles. Now it's line bundle over $\mathbb R P^2$. I know that this bundle is not trivial. So I want list up to equivalence all bundles over $\mathbb R P^2$.

When I've studied bundles over spheres there was great approach that called clutching functions. But if there is some direct methods to do this (like in the link), I'd like to know them.

EDIT

In this post user86418 has constructed continuation of local trivilization. Maybe we can do something similiar here?

Answer 1

Real line bundles are classified by their Stiefel-Whitney classes. Real line bundles over $X$ correspond to $H^1(X;Z_2)$. In case of $RP^2$ there seems to be only 2 line bundles: tautological and trivial.