  We prove that an isometric immersion of a simply connected Lorentzian surface
in $\mathbb{R}^{2,2}$ is equivalent to a normalised spinor field solution of a
Dirac equation on the surface. Using the quaternions and the Lorentz numbers,
we also obtain an explicit representation formula of the immersion in terms of
the spinor field. We then apply the representation formula in
$\mathbb{R}^{2,2}$ to give a new spinor representation formula for Lorentzian
surfaces in 3-dimensional Minkowski space. Finally, we apply the representation
formula to the local description of the flat Lorentzian surfaces with flat
normal bundle and regular Gauss map in $\mathbb{R}^{2,2},$ and show that these
surfaces locally depend on four real functions of one real variable, or on one
holomorphic function together with two real functions of one real variable,
depending on the sign of a natural invariant.
