I am also unclear about the definition of the spin representation. But if it is the exterior algebra of $V=Sym^{N-1}(W)$ with $W$ the two dimensional defining rep, it is enough to know the decomposition of the graded pieces $$ \wedge^k(Sym^{N-1}(W))\simeq Sym^{k}(Sym^{N-k}(W))\ . $$ This is known since around 1856 and is called the Cayley-Sylvester formula (see this MO post). As for the isomorphism above also called the Wronskian isomorphism it is explained e.g. in this article, section 2.5.

I am also unclear about the definition of the spin representation. But if it is the exterior algebra of $V=Sym^{N-1}(W)$ with $W$ the two dimensional defining rep, it is enough to know the decomposition of the graded pieces $$ \wedge^k(Sym^{N-1}(W))\simeq Sym^{k}(Sym^{N-k}(W))\ . $$ This is known since around 1856 and is called the Cayley-Sylvester formula (see this MO post). As for the isomorphism above also called the Wronskian isomorphism it is explained e.g. in this article, section 2.5.