Most general Fock state of spin-1/2 fermions: a parametrization

Question

I wanted to ask if it is true that a valid parametrization of a generic (unnormalized) state in the Fock space of spin-$1/2$ fermions can be written as: $$|\psi>= \prod_{j=1}^L[\alpha_j +\beta_j \hat c^{+}_{j,-}+\gamma_j \hat c^{+}_{j,+}+\delta_j \hat c^+_{j,-}\hat c^+_{j,+}]|0>,$$ where I have introduced the fermionic creation operators at site $j$ with spin $\sigma=\pm$, and the terms multiplying them are arbitrary coefficients. I believe it is not, since it captures only product states. Thanks in advance for the help!