Grassmann manifold $G_{k,n}$ is the set of k-dimensional subspaces of $\mathbb{R^n}$. Let’s consider the set of k-frames $V_{k,n}$. I want to show that $$ G \to V_{k,n} \to^{\pi} G_{k,n} $$ can be given the structure of a principal bundle. First we define a projection $\pi: V_{k,n} \to G_{k,n}$ by taking any k-frame in $V_{k,n}$ to the subspace spanned by that frame. And for any element in $G_{k,n}$ the fiber is the set of all k-frames which are contained in that element in $G_{k,n}$. Furthermore $G$ could be taken as $O(k)$. Could anyone please tell me what are the local trivializations?!
Thanks in advance.