in bordism-theory and many bordering areas one has the following construction: Given a manifold M (say closed for the purposes of this discussion and k-dimensional), we embed it into some $\mathbb R^n$ for n large and look at the normal bundle of that embedding. This pulls back to give an n-k-dim vectorbundle over M, and we consider the homotopy class $M \rightarrow BGL(n-k) \rightarrow BGL$, where the first map is the classifying map for that bundle and the second one is induced by the obvious inclusion. One now finds that the homotopy class of this composition does not depend on the particular embedding chosen. Since $BGL$ classifies principal-$GL$-bundles we have thus constructed an isomorphism class of such bundles and from what I gather this is what is called the stable normal bundle.
Now my question is:
Is there a sufficiently nice construction of an actual $GL$-bundle representing this isomorphism class?
There certainly seems to be none for the individual normal bundles (for they of course DO depend on the embedding for small n), but for the infinite one there just might be, right? By 'construction' I mean construction out of intrinsic data of the manifold and not one along the lines of 'embed M into $\mathbb R^\infty$ and look at the frames of the arising normal bundle'.
If a construction can be found at all then there are probably many, so there won't be a canonical one, which is why I don't really want to specify what 'nice' is supposed to mean.
Thank you for any answers