  In this paper, we investigate the parametric version and non-parametric
version of rigidity theorem of spacelike translating solitons in
pseudo-Euclidean space $\mathbb{R}^{m+n}_{n}$. Firstly, we classify
$m$-dimensional complete spacelike translating solitons in
$\mathbb{R}^{m+n}_{n}$ by affine technique and classical gradient estimates,
and prove the only complete spacelike translating solitons in
$\mathbb{R}^{m+n}_{n}$ are the spacelike $m$-planes. This result provides
another proof of a nonexistence theorem for complete spacelike translating
solitons in \cite{C-Q}, and a simple proof of rigidity theorem in \cite{X-H}.
Secondly, we generalize the rigidity theorem of entire spacelike Lagrangian
translating solitons in \cite{X-Z} to spacelike translating solitons with
general codimensions. As a directly application of theorem, we obtain two
interesting corollaries in terms of Gauss image.
