I agree with Spiro Karigiannis that the question would need some clarifications. I slightly modify the question by replacing the first "form" by "for".
In this case I also agree with Spiro about the fact that a metric would be needed to speak of Hodge decomposition... Unless you refer to the Hodge-Lepage decomposition which has a meaning for a symplectic manifold without metric :

 Given a symplectic vector space $(V,\omega)$, one can associate to it two operators $\omega^+$ and $\omega^-$ which act on the exterior algebra on $V^*$ by respectively left multiplying a given form $\alpha$ by $\omega$ (i.e. $\omega^+(\alpha)=\omega\wedge\alpha$) or by contracting $\alpha$ by the Poisson bivector $\pi$ associated to $\omega$ ($\omega^-(\alpha)=i_\pi(\alpha)$).
These two operators satisfy the relations of the Lie algebra $sl(2)$ and they cut out the space of differential forms into irreducible $sl(2)$-modules which are also modules over the Lie algebra of symplectomorphisms since the operators $\omega^+$ and $\omega^-$ are invariant under the action of this Lie algebra. This decomposition is called the Hodge-Lepage decomposition, and the highest weight vectors are called effective forms. One can find all the details, and some explicit formulas in Darboux coordinates in chapter 5 of the book "Contact Geometry and Nonlinear Differential Equations" by Kushner, Lychagin and Roubtsov (encyclopedia of Mathematics and its Applications (No. 101)) at Cambridge University Press.