I am studying Stiefel-Whitney classes at the moment. In his book
http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf
Hatcher defines orientability of a vector bundle $p:E\to B$ (with path-connected base space $B$) by a map $\pi_1(B)\to \mathbb{Z}_2$ by assigning $0$ if a loop preserves orientation and $1$ if it reverses orientation. If the map is zero for every loop then the bundle is orientable.
(This map can also be seen as an element in $H^1(B,\mathbb{Z}_2)$ and is named the first Stiefel-Whitney class $w_1(p:E\to B)$. Thus we can neatly write $p:E\to B$ iff $w_1(p:E\to B)=0$)
I am concerned now about the orientation definition of vector bundles. We define it usually as Wikipedia does
https://en.wikipedia.org/wiki/Orientation_of_a_vector_bundle
i.e. an orientation of a vector bundle coincides locally with the $\mathbb{R}^n$-standard orientation.
Now why are these two notions equivalent? It is not obvious to me because on the one hand we have loops and on the other hand open sets.
A corollary of this would also be that if the base space $B$ is a CW-complex then $B$ is orientable iff its 1-skeleton is orientable, because every loop can be homotopically deformed to lie in the 1-skeleton.