Why do you want to update BIOS? Are you having problems with the system, stability or compatibility issues? There's a slight risk with updating BIOS and it's not recommended unless system has problems or there have been hardware changes and the update is necessary.
However, if you want to update anyway you don't need a Windows application.
You can format a USB storage stick FAT32. Copy the BIOS file (v2203 in your case) to it's root directory (that means do not put the file in a folder on the USB drive). Turn system off. Plug the USB in. Turn on and enter BIOS by pressing Delete/F2 or whatever is applicable.
Go into the Advanced Mode of the BIOS. Go to the Tool menu to select ASUS EZ Flash 2 Utility and press Enter to enter it. Once...