I interpret the question differently from the other answers and comments. It appears to me that the question refers to the method of moving frames, where one uses differential forms to study Riemannian geometry. Most expositions start with a choice of an orthonormal frame of tangent vectors and work with the corresponding basis of dual 1-forms. However, all of this simplifies, if one works on the frame bundle instead. Instead of having to choose a moving frame, you work with a canonical set of 1-forms defined on the whole bundle that effectively do the calculations in all possible moving frames in one shot. One advantage of this, for example, is that the frames and calculations involving them become global rather than local, when the manifold is orientable.
Unfortunately, I don't know of a reference for this. I learned it from Robert Bryant, and we can hope that he'll answer this question. Otherwise, I'll try to add more details later.