I would like to understand how to construct a vector bundle over the n sphere give a map of its equatorial $(n-1)$-sphere into the general linear group $GL_n(\mathbb{R})$. My thought is that one extends the map to a tubular neighborhood of the equator by projection then uses this as the coordinate transformation between the two hemispheres.
I would then like to compute the top Stiefel-Whitney number of this bundle. There is a paper of Milnor that says the top Stiefel-Whitney number is not zero.