Why is the tangent space of $O(n)$ at $H$ equal to $T_H O(n) = \{ M \in \mathbb{M} ( n, \mathbb{R} ): (DF(H))(M) = 0 \}$, where $$F: \mathbb{M} ( n, \mathbb{R} ) \cong \mathbb{R}^{n^2} \to \mathbb{Sym}(n)\cong \mathbb{R}^{\frac{n(n+1)}{2}}$$ is given by $$F(A) = AA^T$$
Is it a general phenomenon? I mean if we have a map $F$ from a manifold to another manifold, is it possible to find the tangent space of $f^{-1}({c})$ as a manifold provided that it is a manifold, i.e, the derivative is full rank at every point in the pre-image?