Tripwire is a linux util for doing just that. However you need some read-only media to store the hashes and I think rootkits can still just intercept the read calls.
You need to run 'tripwire --update' every time you run 'apt-get update' or 'pip install foo' or 'npm install bah' or whatever - then you wont get that storm of false positives.
I've been in the previous boat: of having it running on a system I've inherited, and given up because it seemed too much hassle.
Those three timely notifications of real breaches have made 20+ years worth of occasional false positives 100% worth it.