Characterizations of Euclidean room

There are probably 3 means of identifying the abstract Euclidean room $E^n$ that are fairly various in spirit :

1. axiomatically (with axioms worrying measurement)
2. by the abstract Euclidean team $E(n)$ (as its proportion team, establishing $E^n$ distinctly)
3. by infering a statistics and also calling for that the room is a topmost one relative to the building that the $(n+1)$ - dimensional Cayley - Menger component disappears for all $(n+2)$ - tuples of factors and also does not disappear for all $k$ - tuples of factors "in basic position" for $k < n+2$ .

^{[Having actually attempted to "rescue" 3 by including problems in italics , as a result of Robin is comment. ]}

Inquiry 1 : Is it proper, in fact, that $E^n$ is distinctly established using 2 and also 3?

Inquiry 2 : What are various other means of identifying $E^n$ that are fairly various in spirit?