Let $\text{Or}_k(\mathbb{R}^n)$ be set of orthonormal $k$-frames for $k\leq n$ (collection of set of orthonormal $k$ vectors in $\mathbb{R}^n$) and let $\text{Gr}_k(\mathbb{R}^n)$ be Grassmannian of dimension $k$. I want to show the map from $\text{Or}_k(\mathbb{R}^n)$ to $\text{Gr}_k(\mathbb{R}^n)$ by sending an lemenet to subspace it spans is topological quotient map. It is known that the set of $n\times n$ orthogonal real matrices $O(n)$ acts transitively on $\text{Or}_k(\mathbb{R}^n)$. So we can equip $\text{Or}_k(\mathbb{R}^n)$ with topology coming from $O(n)/\text{Stabilizers of} \{e_1,\dots,e_k\}$ where $\{e_1,\dots,e_k\}$ is subset of standard basis and the stabilizer is isomorphic to $\{\text{Permutation matrices of order } k\times k\}\times O(n-k)$.
Now we have quotient map $O(n)\rightarrow O(n)/\{\text{Permutation matrices}\}\times O(n-k)\cong \text{Or}_k(\mathbb{R}^n)$. We have also quotient map $O(n)\rightarrow O(n)/(O(k)\times O(n-k))$ and by universal property of quotient, we get unique continuous map from $O(n)/\{\text{Permutation matrices}\}\times O(n-k)$ to $O(n)/O(k)\times O(n-k)$. We can check that the map sending $k$ orthonormal vectors to it spans makes certain diagram commutes, and we need to check this map satisfies property of quotient map. Is this the correct way to get what I want?