See docs here. Note that sys.exception(), within an except, returns the current exception.
import traceback
from datetime import datetime
try:
open("does_not_exist")
except Exception as e:
traceback.print_exc()
with open("exc.txt","at") as f:
print(f"==[ {datetime.now().strftime('%c')} ]==",file=f)
traceback.print_exc(file=f)