Duality for Minors of Symmetric Matrix

Question

Let $G=\textrm{SL}_n(\mathbb{C})$ be the special linear group over the complex numbers. It acts on the set of symmetric $n\times n$ matrices via $g.A:=g\cdot A\cdot g^t$. This action induces an action of $G$ on the vector space $V_d$ spanned by all the $d\times d$ minors (considered as polynomials in the entries of $A$). We clearly have $V_0\cong V_n$ as $G$-modules: Both are the trivial representation. We also have an isomorphism $V_1\to V_{n-1}$ that sends $A$ to $\textrm{tr}(A\cdot \textrm{adj})$ (here $\textrm{adj}$ denotes the adjugate matrix, its entries are $n-1\times n-1$ minors). Does this generalize? Is there a natural $G$-invariant isomorphism $V_d\to V_{n-d}$ and can we describe it explicitely?