  We explore the properties of the SO(3) Majorana representation of spin. Based
on its non-local nature, it is shown that there is an equivalence between the
SO(3) Majorana representation and the Jordan-Wigner transformation in one and
two dimensions. From the relation between the SO(3) Majorana representation and
one-dimensional Jordan-Wigner transformation, we show that application of the
SO(3) Majorana representation usually results in $Z_{2}$ gauge structure. Based
on lattice Chern-Simons gauge theory, it is shown that the anti-commuting link
variables in the SO(3) Majorana representation make it equivalent to an
operator form of compact $\text{U(1)}_{1}$ Chern-Simons Jordan-Wigner
transformation in 2d. As examples of its application, we discuss two spin
models, namely the quantum XY model on honeycomb lattice and the $90^{\circ}$
compass model on square lattice. It is shown that under the SO(3) Majorana
representation both spin models can be exactly mapped into $Z_{2}$ gauge theory
of spinons, with the standard form of $Z_{2}$ Gauss law constraint.
