Let $M$ be a complex manifold of real dimension $2N$, equipped with a Hermitian metric. The corresponding Hodge map $\ast$ has the following properties:

(i) It is a ${\bf C}$-linear map $\ast:\Omega^k(M) \to \Omega^{2N-k}(M)$;

(ii) $\ast(\Omega^{(p,q)}(M)) = \Omega^{(N-p,N-q)}$(M);

(iii) $\ast^2 = (-1)^{k}$ on $\Omega^k(M)$.

Now I would guess that there exist other maps on $\Omega(M)$ with these properties which do not arise as Hodge maps from some Hermitian metric. So my question is, do there exist extra (algebraic) properties of $\ast$, which when put together with $(i),(ii)$, and $(iii)$, determine all the Hodge maps, but without ever explicitly mentioning metrics.