@LeviAckermen Ask the professor. It’s (usually) illegal to do so otherwise. Professors’ lectures are their intellectual property.
This is going to vary from class to class. Some professors may not mind at all, and other professors won’t allow audio/video recording of any kind, unless one has a documented learning disability. Many professors share concerns about intellectual property rights being infringed by recordings.
If you intend to record a class, always be sure to ask the professor for permission prior to recording.
You should ask the permission for video record.
You’ll need to read the policy on recording lectures.
I remember at my community college that there was a strict NO recording whatsoever of lectures stated in the template syllabus that every instructor used. This was college policy. If you were caught recording, you’d be given an automatic F for the class and removed. I always thought it was ironic that the bookstore’s website advertised tape recorders for lectures on the front page as the main page graphic. 
When I was taking an Intro Linguistics course, there was a student next to me recording the lecture on her tablet.
And don’t worry about the characters being sloppy. Even there are native Japanese people who have sloppy handwriting when they take notes. Also, speaking as someone who graduated as a Japanese major, I’ve never taken notes in any of my Japanese language courses.