 Symplectic group as a regular level set

The map $F:M_{2n}(\mathbb{R})\to {\mathrm{Skew}}_{2n}(\mathbb{R})$ given by $F(A)=A^{T}JA$ has $Sp(2n,\mathbb{R})=F^{-1}(J)$. At a symplectic $A$,is surjective, so the regular level set theorem makes the symplectic group an embedded submanifold.