If you finished Free Will, you already defeated Niblet Omega, for the Diviners in Fountain Square. For Skeletons in the Closet, on the other hand, Jamus, your client, will ask you to find and defeat Skeleton Omega, or the last Skeleton in the video game.
Finding Skeleton Omega is not an easy task, but since the payout is high, you should finish Skeletons in the Closet the moment it becomes available.
The following walkthrough explains the steps you need to follow so you can find and defeat the last Skeleton in Lightning Returns: Final Fantasy XIII.
Description: “In Ruffian, Lightning encounters a former bandit by the name of Jamus. He is a tormented soul, who cannot forget the day when Skeletons attacked him and his children and left him as the only survivor. Jamus’s life is in tatters, and the only thing that might save him is revenge. Lightning agrees to seek out all Skeletons and destroy them once and for all.”
Location: Dead Dunes
Requirements: Complete main-quest 4-4: Murals of the Gods
Rewards: 4500 Gil, Maximum HP + 140, Strength + 14, Magic + 12, Pioneer’s Eyepatch Adornment
After you complete main-quest 4-4: Murals of the Gods, return to Ruffian and enter the main building looking for a NPC wearing an eye patch.
Talk to him, and he will ask you to kill all Skeletons in Dead Dunes, along with Skeleton Omega. At this point, if you open the quest-log you should see how many skeletons you have left until Omega Skeleton spawns.
Note: The skeletons you have defeated before activating the quest, also count. You need to defeat 60 Skeletons before Omega Skeleton appears.
If the number of Skeletons you have to kill is too high, you can opt to wait until you gain access to the Ultimate Lair and look for Omega Skeleton on floor 20. On the other hand, if the number is low, you should head to the Temple Ruins and kill the remaining Skeletons.
When the counter reaches 0, the next Skeleton you will fight is the Omega Skeleton. Unlike other Skeletons, the Omega is more powerful because it has four times the health of a normal Skeleton; however, its vulnerabilities are the same.
Fire, ice, air and physical attacks will help you stagger the Omega Skeleton. Most debuffs, including Deshell and Deprotect, also stick.
Once staggered and debuffed, activate the Overclock ability to kill the Last One. Now, head back to Jamus to complete Skeletons in the Closet.