By Liouville's theorem the symplectic volume of a symplectic manifold $(M,\omega)$ is preserved under symplectomorphisms. One usually uses the language of classical mechanics to show that and performs some sort of a canonical transformation which is a symplectomorphism and then shows that the volume is unchanged.
Is there a more formal way to show that without the need to introduce canonical coordinates and canonical momenta, without talking about the phase space as we normally do in classical mechanics and so on? Is there a way to see it in the level of the Hamiltonian flows and the Hamiltonian vector fields? If yes, what is this way and could you provide a reference?
If $(M,\omega)$ is a $2n$-symplectic manifold and $f$ a symplectomorphism, $f^*\omega=\omega$ implies $f^*\omega^n=(f^*\omega)^n=\omega^n$. But $\omega^n$ is the volume form.