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 $Ascher$.

The links for downloading these paper are:

"https://eudml.org/doc/172437"

"https://eudml.org/doc/172514"