  To study spacelike surfaces of codimension two in the Lorentz-Minkowski space
$\Bbb R^{n+1}_1,$ we construct a pair of maps whose values are in
$HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\},$ called $\textbf n_r^{\pm}$-Gauss
maps. It is showed that they are well-defined and useful to study practically
flat as well as umbilic spacelike surfaces of codimension two in $\Bbb
R^{n+1}_1.$
