I want to show that the set of orthogonal matrices, $O(n) = \{A \in M_{n \times n} | A^tA=Id\}$, is an embedded submanifold of the set of all $n \times n$ matrices $M_{n \times n}$.
So far, I have used that this set can be described as $O(n) = f^{-1}(Id)$, where $f: M_{n \times n} \rightarrow Sym_n = \{A \in M_{n \times n} | A^t = A\}$ is given by $f(A) = AA^t$, and that the map $f$ is smooth. Hence I still need to show that $Id$ is a regular point of this map, i.e. that the differential map $f_*$ (or $df$ if you wish) has maximal rank in all points of $O(n)$.
How do I find this map? I tried taking a path $\gamma = A + tX$ in $O(n)$ and finding the speed of $f \circ \gamma$ at $t=0$, which appears to be $XA^t + AX^t$, but don't see how to proceed. Another way I thought of was by expressing everything as vectors in $\mathbb{R}^{n^2}$ and $\mathbb{R}^{\frac{n(n+1)}{2}}$, but the expressions got too complicated and I lost track.