I found a proof of this fact in the paper Tarski's problem and Pfaffian functions by Lou van den Dries published in Logic Colloquium 84.
I was wondering if there are more direct (or more elementary) proofs of this fact because the papers I found they all reference this original one by van den Dries.
Theorem: Let $M$ be a Dedekind complete o-minimal structure. Then every type in $S_n(M)$ is definable over $M$.
When $n = 1$, the result is a fairly straightforward consequence of o-minimality. For a sketch of a proof, see Lemma 2.4.4 of Notes on o-minimality and variations by Dugald Macpherson.
The proof for general $n$ is substantially more difficult. It was originally proved in 1994 in the paper Definable types in o-minimal theories by Marker and Steinhorn (Corollary 2.2). Pillay gave another proof the same year in the paper Definability of types and pairs of o-minimal structures (Theorem 1.1).
You asked for a more direct or more elementary proof, and maybe what I've given you are more abstract or more complicated proofs... I haven't read the proof by van den Dries of the special case over $\mathbb{R}$, but I would be quite surprised if there were a proof over $\mathbb{R}$ that was substantially simpler than the general o-minimal case.