This is something which always baffled me about discussions of affine geometry. I was told "you have no metric available". But it seems obvious that there is always a metric available.
In n-dimensional affine geometry, will "promoting" a linearly independent set of n vectors and a linearly independent set of n linear forms determine a metric?
I am specifically asking about geometry of real number scalars. By a metric, I mean a symmetric non-degenerate bilinear form. The answer seems to obviously be yes. ... But once you start using your metric you are no longer doing affine geometry.
The original statement of my question was intentionally limited because I didn't want to submit a specific method and have that become the subject of discussion. But for the sake of concreteness, here is one method I had in mind.
We can always decree our given linearly independent spanning vector set to be orthonormal. Since affine geometry has no inherent definition of orthogonality, and no means of comparing non-parallel displacements, there is nothing to contradict that choice.
Using our fiat orthonormal basis
$$\left\{ \hat{\mathfrak{e}}_{i}\backepsilon1\le i\le n\right\},$$
we determine those $n$ linear combinations of our given one-forms such that
$$\hat{\omega}^{j}\left[\hat{\mathfrak{e}}_{i}\right]=\delta_{i}^{j}.$$
With vectors expressed as $$\mathfrak{v}=\hat{\mathfrak{e}}_{i}v^{i},$$
and one-forms expressed as
$$\overset{\sim}{w}=w_{j}\hat{\omega}^{j},$$
we define the dual of a vector by
$$\overset{\sim}{v}=\sum_{j}\hat{\omega}^{j}\hat{\omega}^{j}\left[\mathfrak{v}\right]=\sum_{j}\hat{\omega}^{j}v^{j}=v_{j}\hat{\omega}^{j}.$$
We now have the standard Pythagorean metric:
$$\overset{\sim}{v}\left[\mathfrak{v}\right]=v_{j}\delta_{i}^{j}v^{i}=\delta{}_{ij}v^{i}v^{j}$$.
The answer is "yes". Any allowable coordinate system in affine space can be designated as orthonormal. This identifies an equivalence class of coordinate systems and a subgroup of the group of affine transformations having the property that the resulting coordinate system is also orthonormal relative to the original arbitrarily designated orthonormal system.
A different designation of a coordinate system not in that subgroup will identify different relatively orthonormal equivalence class, rendering the members of previously identified equivalence class relatively oblique.
Apparently what distinguishes affine geometry from Euclidean geometry is that there is some particular equivalence class of relatively orthonormal systems which is "true".