  We derive a formalism to express the spin algebra $\mathfrak{su}(2)$ in a
spin $s$ representation in terms of the algebra of $L$ fermionic operators that
obey the Canonical Anti-commutation Relations. We also give the reverse
direction of expressing the fermionic operators as polynomials in the spin
operators of a single spin. We extend here to further spin values the previous
investigations by Dobrov [J.Phys.A: Math. Gen 36 L503, (2003)] who in turn
clarified on an inconsistency within a similar formalism in the works of
Batista and Ortiz [Phys.\ Rev.\ Lett. 86, 1082 (2001)]. We then consider a
system of $L$ fermion flavors and apply our mapping in order to express it in
terms of the spin algebra. Furthermore we investigate a possibility to simplify
certain Hamiltonian operators by means of the mapping.
