There are many things that cause someone to be frightened. A DM may decide that the frightened condition requires a DC 10 concentration check, but it is not one of the ways concentration is broken according to the rules. It's a short list:
- Casting another spell that requires concentration.
- Taking damage.
- Being incapacitated or killed.
If you decide that some other circumstance could qualify, that is also covered in the PHB:
The DM might also decide that certain environmental phenomena, such as a wave crashing over you while you're on a storm-tossed ship, require you to succeed on a DC 10 Constitution saving throw to maintain concentration on a spell.
Unless you are applying house rules (which is perfectly fine), there are only a handful of ways to break concentration. If you are talking about house rules that's fine, you should just state that.