Tread with care. There are organizations where "going around management" gets you flagged as a loose cannon.

It probably won't get you fired outright, but does destroy the goodwill that you will need to make meaningful changes down the road.

Now, if you can do it without rocking the boat (perhaps without management ever finding out) then go for it.

It doesn't hurt to cc management at times or find a way to keep them in the loop on your immediate findings etc. That way it's just a demonstration of 'showing initiative' while still being a team player.