As dear Derek Holt said, the answer is yes. These two references completely solved this problem:
$1)$ "The Faithful Linear Representation of Least Degree of $S_n$ and $A_n$ over a Field of Characteristic 2" by $Wagner$ and $Ascher$.
$2)$ "The Faithful Linear Representations of Least Degree of $S_n$ and $A_n$ over a Field of Odd Characteristic" by $Wagner$ and $Ascher$.