Question: Using the Linux Kali distribution, I was able to get the password from my wifi network using the `airmon-ng` [Aircrack-ng] tool.

However, how can I find out if someone is doing this on my network?

Answer: 
You can't tell if they're doing it because they are scanning your network promiscuously, which means they are just monitoring traffic and not generating any of their own.

You **can** normally prevent this type of attack by using an appropriate level of encryption and passphrase sizes, something such as WPA2-PSK instead of WPA.
Assuming you're using WPA authentication, the best thing you can do is have a strong passphrase. WPA is not vulnerable to the same Fluhrer, Mantin, and Shamir key recovery attack that WEP is vulnerable to, but it's still possible to brute force the passphrase offline by capturing the WPA 4 way handshake, so it all boils down to how good your passphrase is. The longer it is, the better it is.