Euclidean incidence spaces

Structure $$\left$$ is an $$n$$-dimensional Euclidean incidence space iff $$B$$ is ternary betweenness relation associated with $$n$$-dimensional Euclidean geometry. That is, Euclidean incidence space is Euclidean geometry minus the notions of congruences.

Let $$n>2$$. Do I conjecture correctly that $$\left$$ is an $$n$$-dimensional Euclidean incidence space iff:

$$(*)$$ There is a matroid $$\left$$ of rank $$n+1$$. If $$A \subseteq X$$ is a flat of rank $$k: 1 \leq k \leq n$$ then $$\left$$ is a $$k-1$$-dimensional Euclidean space.

I believe this to be the case looking at a way in which Hilbert axioms handle dimensions, with planes and lines axiomized extremely carefully, but axiomization of an entire $$3$$-dimensional space basically boiling down to stating something very similar to $$(*)$$.

If that was, indeed, a case, it would be a rather nice tool for checking whether or not something is a high-dimensional Euclidean incidence space.