Here's my updated AutoHotkey script. Run it once, it will set the game fullscreen windowed shortly after it is launched.
#IfWinActive Skyrim ahk_class Skyrim
SetTimer Exile, 1000
WinGet IDs, List, ahk_class Skyrim
Screen_X = %A_ScreenWidth%
Screen_Y = %A_ScreenHeight%
ID := IDs%A_Index%
If ID not in %Old_IDs%
WinActivate ahk_id %ID%
WinSet, Style, -0xC40000, A
New_IDs .= ID ","
StringTrimRight Old_IDs, New_IDs, 1
Don't ask me about the loop, I just pieced together parts of various AutoHotkey scripts that worked. The one I used previously worked with a keyboard shortcut and caused problems in Windows. This one should only ever touch the Skyrim window.
The game should first be set to windowed mode, and full desktop dimensions by editing SkyrimPrefs.ini located in ~/Documents/My Games/Skyrim:
PS: When you alt tab like this, it's very fast but you may get the windows mouse cursor appearing in the game (I think this issue is also with the default fullscreen mode). The quickest fix for me is to press the tilde key, which activates the console. It's easier then going in the game menu, but it's the same principle: pause the game. So toggle console, alt tab, un-toggle console. Should prevent the Windows mouse cursor from appearing. There are also possible solutions with AutoHotkey for the mouse cursor but I haven't tried those yet.