I would suggest replaceing your standard wheel nuts with larger "track nuts". These are designed for indoor track racing, and have an integral knurled washer. They are very large and will hold the wheel firmly in place. They are best tightened with a ring spanner (US= wrench) rather than an open-ended crescent spanner. A standard workshop length one is sufficient. You can apply some foot presure to torque the nut, but not too much.
You can get track nuts from better bike shops, but take your bike so you get the correct thread type.