What's wrong with my argument to compute the vector bundles on $S^1$?

Question

I have read the solution in Vector bundles of rank $k$ with base $S_{1}$, and I am sure that answer is correct. But I have another approach and I don't know where did I make a mistake. So as circle is covered by real line, and any bundle on real line is trivial as real line is contractible. And if we are given a bundle on circle, say, E, we can always pull it back to be a trivial bundle on $R^1$, say, F. And an inner metric will pull back to an inner metric as well. Now if we choose an orthonormal frame on the fiber of 0 in E. Then it will be pulled back to be an orthonormal frame on the fiber of 0 in F. And by idnetification of the fibers of F(as it is trivial), we can get an orthonormal frame on the fiber of 1 in F which is projected to an orthonormal frame on the fiber of 0 in E again. So actually the geometry is that we let the orthonormal frame move around the circle so that will be another orthonormal frame. Hence we can associated it with an orthogonal matrix. And any orthogonal matrix will give rise to a vector bundle. And after a little computation I find out that two different orthogonal matrices give rise to an isomorphic bundle iff the two orthogonal matrices are conjugate. Hence I come to the conclusion that $Vect_k(S^1)$ is equal to {the conjugacy classes of O(k)} which obviously contradict the answer above. Could you tell me what's wrong with my argument?