macbook screen smeared after cleaning :(

I just tried to clean my black macbooks lcd screen because it was getting quite dirty and since the only i had around that seemed suitable was eye glass cleaner and a lens cloth taking care to lightly wipe the screen. Well i think i shouldnt have even bothered to try to clean as now I have smears all over my screen and I have no idea how to get rid of it.

Does anyone have any ideas on how i can get rid of the smears? any suggestions on what i need to clean the screen in the future?