= Solution
Let $\operatorname{Skew}_{2n}$ be the skew-symmetric $2n$ by $2n$ <matrices> and define
$$
F:M_{2n}(\mathbb R)\to\operatorname{Skew}_{2n},\qquad F(A)=A^TJA.
$$
Then $Sp_n(\mathbb R)=F^{-1}(J)$, and
$$
DF_A(H)=H^TJA+A^TJH.
$$
At a symplectic $A$, put $H=AX$. The <derivative> becomes $X^TJ+JX$. Given skew $K$, choosing $X=-\tfrac12JK$ gives $X^TJ+JX=K$, so the <derivative> is surjective.
The <preimage theorem> makes the level set a submanifold. Since
$$
\dim M_{2n}=4n^2,\qquad \dim\operatorname{Skew}_{2n}=\frac{(2n)(2n-1)}2=2n^2-n,
$$
its dimension is $2n^2+n$.
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