3 accessible dimensions in hyperspace

In short: I have a problem with wrapping my head around the idea of an ever expanding infinitely sized universe as well as the idea of a finite universe. The problem with the latter being that there would be a wall, with the former that infinite space is hard to understand. Now the question: Is there a mathematical model for 3 accessible dimensions , this is what we 'perceive', in a higher-dimensional space that wraps around? Not that higher dimensions are easier to understand, nevertheless it could somehow 'explain' *seemingly* infinite space.

In 2 dimensional space I can imaging a *seemingly* infinite space when thinking of the surface of a sphere. The spherical coordinates

x = r * sin(theta) * cos (phi)
y = r * sin(theta) * sin (phi)
z = r * cos(theta)

obviously describe the surface-coordinates in such a 3d space with only 2 dimensions accessible. A point/line can move on top of the sphere forever and will never be able to leave it since this space 'wraps around'. The facts that for

phi = [ 0, 2 * pi ] = [ 2*n*pi, (2*n+1)*pi ]

theta = [ 0, 1 * pi ] = [ m*pi, (m+1) * pi ]

helps me in understanding the wrapping around of this space in mathematical terms.

Is there an equivalent for 3D space in hyperspace where the 3D space wraps around and the additional dimension can remain inaccessible (to us)?

If you have pointers to some background on this, this may be (!) helpful (somewhat of a 'layman' in deeper math). Also, if we call the above 'spherical coordinates', is there a name for the equivalent in higher dimensional (4D) space?

Regards,
Stefan