Stable normal bundle of a manifold Hi,
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
 A: Perhaps I'm confused about what you are looking for, but haven't you already constructed an "actual" $GL$-bundle in your question?
What I mean is the following.  The usual definition of $GL$ is a direct limit of $GL(m)$'s.  So an element of $GL$ is just an element of $GL(m)$ for some $m$.  Similarly, if $M$ is compact then a $GL$ bundle over $M$ is just a $GL(m)$ bundle over $M$, for some $m$.  As you note in your question, one can construct such bundles by embedding $M$ in $\mathbb{R}^{k+m}$, taking the frame bundle of the normal bundle, and then interpreting this as a $GL$-bundle rather than a $GL(m)$-bundle.  Any two such embeddings of $M$ give isomorphic $GL$-bundles.

In response to pudin's comment below, here's a second construction.  Embed $M$ into $\mathbb{R}^\infty$.  Define a bundle $F$ over $M$ whose fiber at $x$ is frames of the normal bundle of the embedding at $x$ which eventually coincide with the standard framing of $\mathbb{R}^\infty$.  (This is possible because the image of the embedding will lie in some $\mathbb{R}^n \subset \mathbb{R}^\infty$ if $M$ is compact.)  $F$ is a principal $GL$ bundle, where in this case we take $GL$ to be invertible linear maps $\mathbb{R}^\infty \to \mathbb{R}^\infty$ which differ from the identity only on a finite subspace.
