Usually, Euclidean space is introduced by giving a three-dimensional vector space ($\mathbb{R}^3$ for simplicity) and the metric $$d_E\big((x_1,x_2,x_3),(y_1,y_2,y_3)\big)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2}$$ (or instead the scalar product).
I find this a bit unsatisfying because it's not a priori geometrically obvious that distance can indeed be measured in this way. In other words, the above equation should be a theorem, not a definition.
Instead, we could take the Euclidean group $E(3)$ and define $d_E$ to be the only metric whose isometry group $E(3)$ is (or something similar if that isn't well-defined).
But this only helps if we can somehow define (or axiomatically characterize) $E(3)$ without reference to the Euclidean metric or the Euclidean scalar product, in such a way that it is intuitively clear that it is the group of isometries of space. And by isometries I mean (the intuitive concept of) transformations of space which preserve the shape of rigid bodies.
Defining the orthogonal group via matrices and adding translations or explicitly talking about rotations and reflections does not satisfy this requirement. So maybe there is a list of "geometrically obvious" properties of the group $E(3)$ and its action on $\mathbb{R}^3$ that uniquely characterize $E(3)$. Or something similar for $O(3)$.
Of course, this is not a 100% precise question, but I hope someone has thought a bit further in this direction and can give an overview of relevant results or guide me to the literature.