Analogues of Beauville-Bogomolov-Fujiki quadratic form on higher-degree cohomology?

Question

The Beauville-Bogomolov-Fujiki form is a quadratic form $H^{2}(X^{n},\mathbf{C})\rightarrow \mathbf{C}$, where $(X,\rho)$ is a hyperKaehler manifold, $\rho\in H^{2,0}(X)$ being closed and nondegenerate. When restricted to $H^{2}(X,\mathbf{R})$ and normalized so that $\int_{X}\rho^{n}\wedge \bar{\rho}^{n}=1$, the form satisfies the equation $$q(\alpha,\beta)=n\int_{X}\alpha\beta\rho^{n}\bar{\rho}^{n}+\frac{1-2n}{n}\left(\int_{X}\alpha\rho^{n}\bar{\rho}^{n-1}\int_{X}\beta\rho^{n-1}\bar{\rho}^{n}+\int_{X}\alpha\bar{\rho}^{n}\rho^{n-1}\int_{X}\beta\rho^{n}\bar{\rho}^{n-1}\right)$$ (cf. Fujiki Relations and Fibrations of Irreducible Symplectic Varieties, Martin Schwald, Jan 21, 2017.) What I am seeking to find now is an analogue of this form that acts on the degree-$(2n-2)$ real de Rham cohomology of a closed symplectic $(4n+2)$-manifold (again, hyperKaehler), but I'm coming up short. Any algebraic geometers out there have some insider knowledge about this to help point me in the right direction?