The classical icon for "approve" is the check mark. Just search on google images for approve, or also if you accept ( similar to approve ) a question on a stackexchange network you will click on a check mark.
There are many examples of using the check mark to display, something is approved. On the GIT-Visualization Software Stash you use a check mark to mark something as approved.
But in the case of a disapproving, there are several possibilities and I wouldn't say, the cross is the classical variant.
Well used is the "Thumb down" icon.
Often you also will see the icon for "deny", while IMO, you should use this to deny something and not really to disapprove.
Also your cross is a good solution
or just mark something red (unuseful in your case).
So to answer to the title: YES, the check mark is the most used (but not official I think) version. For the disapprove icon, I can't say, there is a (in-)official version. IMO, the cross is easy-to-understand and is good practice for use as an icon.