How to set up $N$ symmetric random variables whose sum is always zero

Let $A$ be an $N \times N$ matrix:

$A=\begin{bmatrix}-(N-1)&1&1&\cdots&&&\\1&-(N-1)&1&\cdots&&&\\&&&\ddots&&&\\&&&\cdots&1&-(N-1)&1\\&&&\cdots&1&1&-(N-1)\end{bmatrix}$

Let $X_i$ be i.i.d. $N$ random variables.

Then the $N$ random variables defined by $Y \equiv A \cdot X$ is the answer.