Consider the set $$S = \left\{(x_1, x_2, \dots, x_n) \in \mathbb{R}^n \;:\; \sum\limits_{i = 1}^n x_i = 0\right\}$$
I can see that in $1$D, we just have $x_1 = 0$.
In $2$D, we have the line $x_2 = -x_1 \Leftrightarrow y = -x$.
How to visualize in $3$D? Is there a name for this space?
I think what you're looking for is a hyperplane.