I work on a problem where I'm asked to proove that $SO(n)$ is a submanifold of $R^{n^2}$. If done that by defining a function $f(A)=AA^T-Id$ which satisfies the condition that $f^{-1}(0)=SO(n)$. I procceded by checking that $df(B)=B*A^T+A*B^T$ is surjective as I can find for every $C$ which is element of $Im(f)$ an $B=$$1/2$ $CA$ which 'hits' C. So it seems to me that everything necessary is done, but I haven't used the fact that $detA=1$ for every element of $SO(n)$. Could someone explain to me wether I've done everything correct?
Thank you in advance.