In The Elements, Euclid establishes a set of axioms that dictate the behaviour of the plane. Hilbert refined these axioms to actually prove every proposition of The Elements formally. In Hartshorne's Geometry: Euclid and Beyond, the author shows that any Hilbert Plane satisfying $P$ (the Parallel Postulate) and $D$ (Dedekind's Axiom) is isomorphic to the real cartesian plane. This basically allows us to use analytical methods of geometry in the plane and know that we are fully encapsulating every truth about the plane.
How do we extend this to $n$-dimensional Euclidean space? Are there axioms of space that we want to satisfy which justify our use of ordered tuples of real numbers? Intuitively it seems right. But I'm looking for a rigorous justification. In Hartshorne, he omits an axiomatization of three dimensional Euclidean space. Does one exist?
Hilbert's Foundations of Geometry provides an axiomatization for 3 dimension euclidean space based on points, lines and planes.
Tarski's geometry is based on a single sort (Points) and is modular wrt. dimension: on can choose the lower and upper dimension axioms. On can find the statements for the lower and upper dimension axioms page 182 of Tarski's system of geometry by Tarski and Givant (http://www.geographicknowledge.de/SeminarLFG/TarskisSystemofElementaryGeometry.pdf). For a systematic and very rigorous development of geometry based on these axioms see the book Metamathematische Methoden in der Geometrie.