n-dimensional space:

R = R^1 := the

set of all

real numbers.

R^(n+1) := R*R^n for all

positive natural numbers n, where * is the

cartesian product operator.

One thing to note is that by this definition, the set R^3 actually consists of an ordered pair, the first element of which is a real number, and the second element of which is an element of R^2 (i.e., an ordered pair of real numbers).
Thus, an element of R^3 would be of the form (a,(b,c)). For convenience, however, we allow this to be written instead as an ordered triple, (a,b,c), and analagously for further dimensions.

Thanks to ariels for pointing this out.