Dimension theory "based on $\mathbb R^n$" This question is somewhat vague, so please be gentle with me. I want to know if there is some definition of topological dimension that has $\mathbb R^n$ as a "paradigm", something like 'A nice (connected, normal, whatever) topological space $X$ has dimension $n$ if $X$ can be "related" to $\mathbb R^n$ in some topological way'. Of course, the core of this tentative definition is what we mean by "related". An (almost certainly doomed) attempt of mine, illustrating my point of view (and my intentions) is the following:


  $X$ has topological dimension $n$ if $X$ is homeomorphic to some subspace of $\mathbb R^n$, and it is not homeomorphic to any subspace of $\mathbb R^{n-1}$.


EDIT

Murphy's Law applied in this case. I refrained to ask for a definition not limited to topological manifolds, because I thought that it was clear that my intention was to get an answer with enough generality, for example for spaces like $\mathbb Q^n$. Unfortunately this was not the case. Summarizing, I am asking for a definition not limited to topological manifolds.  Also, please don't forget that the tentative definition above is just for illustration, I am not taking it seriously.
Perhaps what you are aiming at is the more general Urysohn dimension theory, see http://en.wikipedia.org/wiki/Inductive_dimension
The path to connect the space $\mathbb R^n$ with the euclidean space of $n$ dimensions, is through the cartesian product of real number lines.  For this kind of space, one can directly used established 'analytical geometry' formulae (eg a line as $y=ax+b$), to do this.

The relation to non-euclidean space is that it is possible to map the coordinate system $\mathbb R^n$ onto something like hyperbolic space, and then show that there is a topological bending of euclidean into hyperbolic space, and thus some kind of local connectivity applies.

One can consider other sets like $\mathbb C^n$ onto the 'unitary space'.  The 'unitary line' is the argand diagram $\mathbb C^1$.  The same formulae, like the line as $y=ax+b$ still work but the real-line as line is now replaced by the argand-diagram as line.  It's very useful for exploring even dimensions, like $n=4$.  

For example, in 2D, one can represent rotation around a point by the relation $X(t)=X(0) \operatorname{cis}(\omega t)$.  In the 'four-dimensional' $\mathbb C^2$, the same product, will cause a rotation that rotates everything around the point $(0,0)$, specifically:

$$X(t),Y(t) = X(0) \operatorname{cis}(\omega t), Y(0) \operatorname{cis}(\omega t)$$

The cute thing to note here is that ratio $Y(t)/X(t) = Y(0)/X(0)$, and must stay on the same 'argand diagram' or complex line through $(0,0)$.  This equation then describes rotation of all four-dimensional space around a point, and it's relatively easily shown.  Likewise, introducing $Z(), \dots $, show that this is true of all even dimensions.

Mapping things like $\mathbb Q^n$, also is useful.  $\mathbb Q$ is what i call a discrete set, in as much as there are numbers in its scope that don't belong to it: for example, $\sqrt{2}$ is not rational.  A set of this nature is rather like a sparse lattice formed by $\mathbb Z^n$.  For example, the set $CQ$ contains units on the unit circle like $0.6+0.8i$.  

More usefully, I use it of the sets of the span of chords of a polygon $\mathbb Zp$, and their corresponding cyclotomic numbers, $\mathbb {CZ}n$.  This leads to spaces like $\mathbb Zp^n$.

The uniform polytope 'bi-truncated 24choron', or 'octagonny', or $o3x4x3o$, tiles space in a piecewise discrete manner.  This means that we can completely make all of the figures that are attatched to it.  One can evaluate the space that can be reached by edge-traversal, rather like the cubic gives $\mathbb Z^4$.  In this case, it's $\mathbb Z4^4$, the set of coordinates of the form $z_1 + z_2 \sqrt{2}$.  This space is infinitely dense, but is itself a double-projection of something in 8-dimensions, like $E_8$.  There's a set of quaterions based on octagonny.