The symplectic group is defined as the group of matrices $X$ such that $X^T J X = J$ for some fixed skew-symmetric matrix $J$.
Why does $J$ have to be skew-symmetric? Verifying the group axioms doesn't depend on the skew-symmetry of $J$, and any group defined the same way except with an arbitrary matrix $J$ is well-defined, in that it's always non-empty since the identity satisfies the condition trivially.
So why do we specify that $J$ should be skew-symmetric?