Lie algebra of invariant polynomials or invariant smooth functions

Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those smooth functions $f:M_{2n}(\mathbb{R})\to \mathbb{R}$ with $f(AB)=f(BA)\;\;\;\; \forall A,B \in M_{2n}(\mathbb{R})$, would be closed under the Poisson bracket ?

If I understood correctly a rephrasing of question is requiring a symplectic structure such that the set of commutators $AB-BA$ is contained inside a coisotropic submanifold. It shouldn't be difficult to find a symplectic structure on the space of matrices such that the linear subspace $\mathfrak{sl}_n$ (which contains all commutators, being linearly generated by it) is coisotropic.