= Solution
Use the decomposition into <Lagrangian subspaces> $V=V_+\oplus V_-$ from part (a). The <bilinear form> $\Omega$ pairs $V_+$ and $V_-$ nondegenerately: a <vector> in either summand annihilating the other already annihilates its own summand and hence all of $V$, so is zero.
Choose a <basis> $e_1,\ldots,e_n$ of $V_+$. The <linear map>
$$
V_-\longrightarrow V_+^*,\qquad f\longmapsto\bigl(e\longmapsto\Omega(e,f)\bigr)
$$
is an <linear isomorphism>. Let $f_j$ correspond to the <dual basis> of the $e_i$. Then $\Omega(e_i,f_j)=\delta_{ij}$, and the pairings within each summand vanish. Thus
$$
\boxed{(e_1,\ldots,e_n,f_1,\ldots,f_n)\text{ is a symplectic basis},\quad Se_i=e_i,\quad Sf_i=-f_i.}
$$
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