The basic idea is: I would like to draw over everything on the screen.

One way I can imagine this is creating a transparent full-screen window without window controls (minimise, maximise, etc.) or borders. Then drawing into that window which is transparent. The problem I can think of is that I will be unable to control windows which are behind our transparent window.

How could I do something similar to this, without the mentioned problem? I would also like it to work on multiple operating systems if possible.

Edit:
The user will not be drawing with the mouse or other means on the screen, but will be able to continue use his desktop like normal, without that my program interferes in any way (other than the drawing on the screen). My program will only display something on the screen, which the user will be unable to interact with (at least that's the plan).