How would I start off proving that $S= $(the set of symmetric $n\times n$ matrices) is a manifold. I tried using the definition directly by saying $M_n =$ the space of all $n\times n$ matrices For every $A\in M_n$ there exists open sets $U=V=M_n$ and a bijection $F: U\to V$ by $F(A)= A-A^T$ Therefore we have $F(U \cap S) = F(S)$ since $S$ is a subset of $M_n=\{0\} \cap M_n$ this is where I get stuck. Also, I know that the set of all symmetric $n\times n$ matrices is $(n^2 + n)/2$, therefore that is the dimension of the manifold
Definition: A set $M$ (subset of $\Bbb{R}^n$) is a $k$-dimensional manifold if for every $x\in M$ there exists open sets $U$, $V$ and a bijection $h:U\to V$ with $x\in U$ and $H(U \cap M) = V \cap (\Bbb{R}^k \times \{c^{k+1},\ldots ,c^n\})$ for all $c$'s constants