Does the $O(n)$ bundle of a manifold depend on the metric?

Question

Let $g_1$ and $g_2$ be two Riemannian metrics on a manifold $M$. These induce two $O(n)$ bundles on $M$, whose fibers over each point $x\in M$ are the groups of orthogonal transformations of $T_x M$ with respect to $g_1$ or $g_2$, respectively. Are these two $O(n)$ bundles necessarily isomorphic?