This is an extremely elementary question but I just can't seem to get things to work out. What I am looking for is a natural definition of the quotient bundle of a subbundle $E'\subset E$ of $\mathbb R$ (say) vector bundles over a fixed base space $B$. Every source I find on this essentially leaves the construction to the reader. I would like to glue together sets of the form $U\times E_x/E'_x$ where $x\in U$ is a locally trivial neighborhood by some sort of transition function derived from those corresponding to $E$ and $E'$, but this doesn't actually make sense in any meaningful way.
While I am tagging this as differential geometry, I would like a construction that works in the topological category (i.e., does not invoke Riemannian metrics) and avoids passing to the category of locally free sheafs.
Sorry if this is a repost (I'm sure it is, but I can't seem to find anything) and thanks in advance.