Centers of higher degree forms

Let Θ be a symmetric d-linear form on a vector space V of dimension n over a field k. Its center, Cent(Θ), is the analog of the space of symmetric matrices for a bilinear form. If d>2, the center is a commutative subalgebra of End(V). It seems difficult to determine which subalgebras can be realized as Cent(Θ) for some some d-linear form Θ. As a first step we conjecture that the center has dimension at most n. The conjecture is proved for n 5.