As well-known, $O(n)$ can be defined as a codimension $(n^{2}+n)/n$ submanifold of $\mathbb{R}^{n^{2}}$ via $A \mapsto A^{t}A$. Another way of constructing $O(n)$ as a manifold is construct smooth chart directly; by defining $M_{n}(\mathbb{R})^{\sharp}:= \{A \in M_{n}(\mathbb{R}): \det(I+A) \neq 0 \}$ and use $O(n)\cap M_{n}(\mathbb{R})^{\sharp}$ and its translation with Cayley transform $A \mapsto A^{\sharp}:=(I-A)(I+A)^{-1}$ as chart map.
My question is, these two ways of construction gives the same manifold structure? How can I show these are the same manifold or not?