An $SU(n)$ structure is *not* a collection of charts whose transition maps have derivatives valued in $SU(n)$. It is a collection of bases of tangent spaces, forming a principal $SU(n)$-subbundle of the bundle of all bases of tangent spaces. The forms you have specified uniquely determine (by pure linear algebra) a collection of bases of tangent spaces: the bases which are identified by linear isomorphism with the standard basis of $\mathbb{C}^n$ in such a way that these tensors are identified with the standard $SU(n)$-invariant tensors in $\mathbb{C}^n$. For example, you might look at Dominic Joyce's book Riemannian Holonomy Groups and Calibrated Geometry, Oxford University Press, pages 36 to 39, for more information. In fact, $SU(n)$-structures depend locally on $3n^2-1$ functions of $2n$ variables. The objects you were thinking of, given by transition maps with derivatives in $SU(n)$, are much more rigid, and are equivalent to flat $SU(n)$-structures. These are locally unique up to diffeomorphism and complex unimodular affine rigid motion of $\mathbb{C}^n$, so if we don't divide out by the diffeomorphism pseudogroup action, then flat $SU(n)$-structures depend on $2n$ functions of $2n$ variables locally.

assumethat $M$ has an almost complex structure, the form $\Omega$ satisfying the conditions (i) and (ii)definean almost complex structure on $M$, namely, the one for which $\Omega$ has type $(n,0)$. – Robert Bryant Mar 19 '13 at 19:08