Consider the function $F:Mat_n → Sym_n$ defined by the formula $F(A) = A∗A$. $Mat_n$ denotes the vector space of n × n matrices with real entries, while $Sym_n$ denotes the vector space of symmetric $n × n$ matrices with real entries. We will also identify $Mat_n$ with $\mathbb{R^n}^2$, and $Sym_n$ with $\mathbb{R^\frac{n(n+1)}{2}}$
I have a general layout of the proof, but no idea how to actually implement it. First, I should show that for every $A ∈ Mat_n$, the differential map $DF(A)$ is given by $[DF(A)](B)=A∗B+B∗A$. Next, I should prove that for every invertible matrix $A∈Mat_n$, one has $dim(ker(DF(A)))=\frac{n(n − 1)}{2}$. From all this, I should conclude that the orthogonal $n×n$ matrices form a $C^1$ surface of dimension $\frac{n(n − 1)}{2}$ in $\mathbb{R^n}^2$.