This is perhaps a bizarre and somewhat philosophical question but nevertheless, I find it a bit confusing.
When we define $\mathbb{R}^n$ with its canonical Riemannian metric, one can go about it like this:
Any smooth manifold starts off as a set so does $\mathbb{R}^n$. To show it is a topological manifold, it is certainly enough to consider the identity map. We may then construct a smooth atlas just out of this chart and thus it is a smooth manifold.
Thus we have the tensor bundles. Since the coordinates are global, they are all trivialized by structures arising from the single chart. Thus any Riemannian metric on $\mathbb{R}^n$ can be written
$$g= g_{ij} dx^{i} dx^{j}$$
for some smooth functions of the canonical coordinates $x^{i}$. At this point one may define the canonical Riemannian (or Lorentzian) metric by requiring $g_{ij} = \delta_{ij}$ or the Lorentzian analogue.
Yet somehow this just doesn't feel geometrical in a way; I'd like to review how an elementary notion of length on the inner product space $(\mathbb{R}^n, < \cdot , \cdot>)$ corresponds to the above trivial Riemannian geometry - particularly in the Lorentzian case.
Of course we can't use geodesic normal coordinates as we are trying to define the metric.