I am looking for a surjective map

Where

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 is homeomorphic to

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

Thanks for any input