What is the principal bundle structure of $O(n)$?

Question

Consider the map $\pi:O(n)\rightarrow G(k,n)$ which maps $A\in O(n)$ to the subspace of $\mathtt{R}^n$ spanned by the first $k$ columns of $A$. Here $G(k,n)$ is the Grassmannian manifold. My question is what is the local trivialization of this principal bundle? I guess the fibre is $O(k)\times O(n-k)$. Is it correct? Thanks.