In other words, you ask whether every finite metric space may be isometrically embedded to Euclidean space $\mathbb{R}^n$. Not every. A necessary and sufficient condition is given by the non-negativity of the so called Caley--Menger determinants.

In other words, you ask whether every finite metric space may be isometrically embedded to Euclidean space $\mathbb{R}^n$. Not every. A necessary and sufficient condition is given by the non-negativity of the so called Cayley--Menger determinants.