As dear Derek Holt said, the answer is yes. These two references are related to this problem:
$1)$ "The Faithful Linear Representation of Least Degree of $S_n$ and $A_n$ over a Field of Characteristic 2" by $Wagner$$Ascher$.
$2)$ "The Faithful Linear Representations of Least Degree of $S_n$ and $A_n$ over a Field of Odd Characteristic" by $Wagner$$Ascher$.
The links for downloading these paper are:
"https://eudml.org/doc/172437"
"https://eudml.org/doc/172514"