$\DeclareMathOperator{\rank}{rank}$ I have problem with following task:
Let $f: SO(n) \to S^{n-1}$ will be a function that takes first column of the matrix. Proof that $f$ is regular in every point $M \in SO(n)$.
So I know that elements from $GL(n,\mathbb{R})$ can be treated as elements from $\mathbb{R}^{n^2}$ and that $f$ is projection on first $n$ coordinates. So I need to construct Jacobi matrix $J_f$ and proof that $df$ is epimorphism (equivalently that $\rank J_f = n-1$). Moreover $SO(n)$ is $\frac{n(n-1)}{2}$-dimensional space so $J_f$ will be matrix $n \times \frac{n(n-1)}{2}$. But here I don't know how to formally justify regularity of $f$. First columns do not have to be orthogonal to one another and I don't know if I will always find $n-1$ lineary independent columns. I would be grateful for any tips.
Note that $f$ is of constant rank, as $f(g h)= g.f(h)$ where $g$ is a diffeormorphism (the linear map $g$. As it is surjective it must be of maximal rank everywhere. You can alos check that its derivative near he origin is the map $A_n\to R^{n}$ which maps an antisymmetric matrix to its first column is onto the space of vectors $\perp$ to $(1,0,..0)$.