Diagnosis
Good on you for being careful. That matte anti-glare coating can be damaged by harsh chemicals or rough cloths, leading to smudges or a hazy look.
How to Fix It
Here's the safe way. First, power off the monitor and let it cool. Use a dry, soft microfiber cloth—like the kind for cleaning glasses. Gently wipe the screen in a circular motion. For tougher spots, dampen a second microfiber cloth with distilled water (not tap water, to avoid mineral spots). Wring it out completely so it's just barely damp. Wipe the screen, then immediately follow up with the dry cloth to buff it streak-free.
If That Didn't Work
For persistent grime, you can use a monitor-specific cleaning spray. Spray it onto the dry cloth, never directly onto the screen, to avoid liquid seeping into the edges. Make sure the spray is labeled as safe for anti-glare coatings.
Heads Up
Never use window cleaner, alcohol, ammonia, or abrasive cloths like paper towels. They will permanently damage the anti-reflective coating, and that damage isn't covered under warranty.