  We present a classification of bilinear Majorana representations for spin-$S$
operators, based on the real irreducible matrix representations of SU(2). We
identify two types of such representations: While the first type can be
straightforwardly mapped onto standard complex fermionic representations of
spin-$S$ operators, the second type realizes spin amplitudes $S=s(s+1)/4$ with
$s\in\mathbb{N}$ and can be considered particularly efficient in representing
spins via fermions. We show that for $s=1$ and $s=2$ this second type
reproduces known spin-$1/2$ and spin-$3/2$ Majorana representations and we
prove that these are the only bilinear Majorana representations that do not
introduce any unphysical spin sectors. While for $s>2$, additional unphysical
spin spaces are unavoidable they are less numerous than for more standard
complex fermionic representations and carry comparatively small spin
amplitudes. We apply our Majorana representations to exactly solvable small
spin clusters and confirm that their low energy properties remain unaffected by
unphysical spin sectors, making our representations useful for
auxiliary-particle based methods.
