Get the Gallery key
The key for the Gallery room is needed before you can get the key for room 003. You can get the Gallery key after completing the control room puzzle which unlocks areas previously blocked by water. The key will be at the bottom of the Aqua ring.
Go to the Gallery room
The Gallery room is present in the long corridor which is located in the south of the map near the safe room. Get inside the Gallery room after using the key.
Get the insecticide
The insecticide can be found in the hands of the dead man inside the Gallery room. Grab the pesticide then leave the gallery room.
Use the insecticide
Right after you leave the gallery room head down the hallway past room 002 until you find a big hole in the wall at the end of the corridor. Use the insecticide there.
Go back to the gallery room
Go back to the Gallery room. You will now be able to access the corridor that was previously blocked by the bee hive which you just destroyed. (See How to get the gallery key in Resident Evil remastered?)
Grab the 003 Key
You will find a key at the end of the corridor. Grab the key then examine it and it will say 003 key. Note that room 003 will be right next to you. (See How to get the emblem key in Resident Evil remastered?)