Here's the method I proposed in the comment. Pick a random point $(x_1,\ldots,x_{n-1})$ from the $n-1$-dimensional hypercube. (This amounts to choosing $n-1$ real numbers uniformly in $[0,1]$)
Now, if $\sum_{i=1}^{n} x_i \geq 1$ throw the point out and start again, if not keep it.
Transform the point using the following affine map:
$$f:\mathbb{R}^{n-1}\to\mathbb{R}^{n}: (x_1,\ldots,x_{n-1}) \mapsto (x_1,\ldots,x_{n-1},1-\sum_{i=1}^{n} x_i) \; .$$
Now, similarly to what Fabian does, select $n$ signs $\sigma_i=\pm 1$ randomly and multiply each component of the vector you obtained by these signs.
As pointed out already by joriki and Gerben, for high dimensions, this method is very wasteful since a fraction $\frac{(n-1)!-1}{(n-1)!}$ points will have to be thrown out.