This is probably a very easy question for those familiar with the arithmetic theory of abelian varieties because it is either possible or impossible for "trivial reasons".

Let $A$ be an abelian variety over a number field $K$, say principally polarized if needed later. I want to somehow describe what a full level $n$ structure ($n\geq 4$) on $A$ is in terms of the Galois representation(s) associated to $A$. Is this even possible?

To give a full level $n$ structure is by definition equivalent to giving an isomorphism of group schemes $$A[n] \to (\mathbf Z/n\mathbf Z)^{2g}.$$