Sure, lots of ways of handling this. A script is one option, of course, and not that hard. You could make it ping once a minute and if the ping fails, send an email. That's not hard, especially in BASH.
Commonly this is handled by systems like Zabbix, Zenoss, Nagios or Spiceworks. They all have that functionality. If you lack that internally it might be well worth looking at implementing something more broadly since you are already addressing the need for this one specific thing.