Given are the coordinates of a regular $n$-simplex in $\mathbb{R}^n$ where all edge lengths are 1. For symmetry the vertices are centered on the origin.
The volume of that unit $n$-simplex is $V_n=\frac{\sqrt{n+1}}{n! \sqrt{2^n}}$ (see Wikipedia for definitions and formulas).
As the $n$-simplex is centered we can multiply the coordinates by a factor $\alpha_n$ without changing symmetry. The factor only affects the volume. By which factor $\alpha_n$ the coordinates of an $n$-simplex have to be multiplied to get $V_n=1$?
The first (approximate) values are $\alpha_1=1, \alpha_2\approx1.5194, \alpha_3\approx 2.0397, \alpha_4=2.5598$.
The volume scales with the $n$th power of the scale factor, so $\alpha_n$ is the reciprocal of the $n$th root of the volume you know.