A spacelike constant angle surface in $E_1^3$ (Minkowski 3-space) is defined by its unit normal vector field forms a constant angle with some fixed vector direction. In this paper, we use the Frenet frame to express the surface and construct a family of spacelike constant angle surfaces which possess the given curve as isoparametric curve. The normal vector of each surface forms a constant angle with a fixed vector lying in the spacelike cone and in the timelike cone respectively. Finally, we give some representative examples.
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