The smallest degree faithful representation of $S_n$ is $n-1$. A reference can be found just before theorem 40 in http://arxiv.org/pdf/1107.1413.pdf

I believe the result is due to Burnside.

The smallest degree faithful representation of $S_n$ is $n-1$ in characteristic does not divide $n$. It is Theorem 22 of Chapter 19, Section 8 of Y. G. Berkovich, E. M. Zhmud; Characters of finite groups. Part 2. Translated from the Russian manuscript by P. Shumyatsky, V. Zobina and Berkovich. Translations of Mathematical Monographs, 181. American Mathematical Society, Providence, RI, 1999.

The result is due to Burnside. The proof is inductive.

Dickson proved of the characteristic divides n and n is at least 5, then n-2 is the smallest degree faithful representation.