To describe multiple fermionic particles, we introduce a Fock space $$\mathcal H_F=V_{\alpha=1}\otimes V_{\alpha=2}\otimes \ ...$$ such that each $V_\alpha$ is a two dimensional vector space labelled by an index $\alpha$ that refers to a complete set of quantum numbers. The corresponding ladder operators for each such space can be written as $$a_\alpha=\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}, \quad a_\alpha^\dagger=\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$$ acting on a basis $\{|0\rangle_\alpha=(1,0)_\alpha^T, \ |1\rangle_\alpha=(0,1)_\alpha^T\}$. This works fine as $a_\alpha^\dagger$ sends $|0\rangle_\alpha$ to $|1\rangle_\alpha$, $a_\alpha$ does the opposite, and $a_\alpha^\dagger|1\rangle_\alpha=a_\alpha|0\rangle_\alpha=0$, so that the number of particles for each $\alpha$ can only be $0$ or $1$. It can then be checked that the various anticommutation rules $\{a_\alpha,a_\alpha^\dagger\}=1$, $\{a_\alpha,a_\alpha\}=\{a_\alpha^\dagger, a_\alpha^\dagger\}=0$ hold.

The only remaining problem is that for $\alpha\ne \beta$, there is still symmetry under the transformation $\alpha\leftrightarrow \beta$ when creating particles, against the antisymmetric nature of fermions: we wish instead to have operators such that $$b_\alpha^\dagger b_\beta^\dagger|\text{state}\rangle=-b_\beta^\dagger b_\alpha^\dagger|\text{state}\rangle.$$ I've been told that this can be accomplished by defining $b_\alpha^\dagger=a_\alpha^\dagger \eta_\alpha$, where $$\eta_\alpha=\prod_{\gamma=1}^{\alpha-1}\begin{pmatrix}-1 & 0 \\ 0 & 1 \end{pmatrix}_\gamma.$$ However, I don't see how this works or how it was derived. Could someone provide an explanation?