Such a fibre bundle does not exists if you suppose that it is endowed with a differentiable structure. Stewart has shown that the group of diffeomorphisms of $R^n$ retract to $O(n)$. So every $Diff(R^n)$-bundle has an $O(n)$-reduction.
Stewart, T. E. (1960). On groups of diffeomorphisms. Proceedings of the American Mathematical Society, 11(4), 559-563.