Interpretation of k-particle-spinor-state

Let h=L2(X,S)\mathfrak{h}=L^2(X,S) be the Hilbert space of a single spin particle, where SS is spinor space (i.e. a finite-dimensional Hilbert space). Then for fermions the nn-particle space is given by the Hilbert space ⋀kh\bigwedge^k\mathfrak{h}.

If S=CS=\mathbb{C} (i.e. the spin-less case), then an element ϕ∈⋀kh\phi\in\bigwedge^k\mathfrak{h} can be interpreted as a totally anti-symmtetric function ϕ:Xk→C\phi:X^k\to\mathbb{C}, for example (ϕ1∧⋯∧ϕk)(x1,…,xk):=1k!det(ϕi(xj))i,j=1,…,k.(\phi_1\wedge\cdots\wedge\phi_k)(x_1,\ldots,x_k):=\frac{1}{k!}\det\left(\phi_i(x_j)\right)_{i,j=1,…,k}.

However, if \dim S>1\dim S>1, then what is the common interpretation of an element \phi\in\bigwedge^k\mathfrak{h}\phi\in\bigwedge^k\mathfrak{h}? All that I can come up with is generalize the above and interpret it as a totally anti-symmetric function \phi:X^k\to\otimes^k S\phi:X^k\to\otimes^k S, i.e. \left(\prod_{i=1}^k\phi_i\otimes s_i\right)(x_1,\ldots,x_k):=\frac{1}{k!}\sum_{\sigma\in S_k}(-1)^\sigma\left(\prod_{i=1}^k\phi_{\sigma(i)}(x_i)\right)s_{\sigma(1)}\otimes\cdots\otimes s_{\sigma(k)}.\left(\prod_{i=1}^k\phi_i\otimes s_i\right)(x_1,\ldots,x_k):=\frac{1}{k!}\sum_{\sigma\in S_k}(-1)^\sigma\left(\prod_{i=1}^k\phi_{\sigma(i)}(x_i)\right)s_{\sigma(1)}\otimes\cdots\otimes s_{\sigma(k)}.
What is kind of strange of this interpretation is that I have the feeling that for \dim S >1\dim S >1 not every totally-antysymmetric function X^k\to\bigotimes^k SX^k\to\bigotimes^k S is obtained in this way, i.e. \bigwedge^k\mathfrak{h}\bigwedge^k\mathfrak{h} is not isomorphic to \operatorname{Alt}_k(L^2(X^k,\bigotimes^k S))\operatorname{Alt}_k(L^2(X^k,\bigotimes^k S)).

Did I get something wrong? Or is there a more easy approach to fermionic k-particle spinor states?

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