I am looking for a surjective map

$\displaystyle g:\{S^{n-1}\times [-1,1] \} \to S^{n}$

Where $\displaystyle S^n=\{ x \in \mathbb{R}^{n+1}:||x||=1 \}$

I have figured out the case for n=1(The easy one)

It is basically converting to sphereical coordinates.

I can't find a closed form for it(I'm not even sure if one exists) for general n.

Just for context I'm trying to show that the suspension of $\displaystyle S^{n-1}$ is homeomorphic to $\displaystyle S^n$

So I need the function g above to be a quotient map.

Thanks for any input