Exterior powers of (half) spin representations

Question

Let $S^+$ (resp. $S$) be the positive half spin representation (resp. spin representation) of $Spin(2n)$ (resp. $Spin(2n+1)$). Their isomorphism classes are elements in the representation rings \begin{align*} R(Spin(2n))&=\mathbb{Z}[V, \bigwedge\nolimits^2V, \cdots, \bigwedge\nolimits^{n-1}V, S^+]\\ R(Spin(2n+1))&=\mathbb{Z}[V, \bigwedge\nolimits^2V, \cdots, \bigwedge\nolimits^{n-1}V, S] \end{align*} where $V$ is the standard representation. How can one express any higher exterior powers of the (positive half) spin representations as polynomials in terms of the generators of the representation rings listed above?

Answer 1

Use a computer program. For instance, here is the decomposition into irreducibles of the 6th exterior power of the half-spin representation of Spin(12). To me the result does not look sufficiently regular to make a guess about a closed formula that would describe the decomposition in general. Writing an expression as polynomial in the fundamental representations (I suppose with multiplication meaning tensor product) would seem even harder still.