Can I get rid of my unwieldy keychain now that I have the Skeleton key?

Solution 1:

Unlike a lockpick or the Skeleton Key, a key unlocks the relevant door/container instantly, without you having to play the lockpicking minigame. Moreover, some doors/containers can only be opened with their key. Overall, I doubt you would save much time by stashing your keys somewhere, and if you do you'll run the risk of not having the necessary key on hand for the few locks that require keys.

If you're playing on PC, you can use Darnified UI mod to collapse item groups in the inventory, hiding keys from view.

Solution 2:

As per the wiki:

The Skeleton Key is an unbreakable lockpick that increases your Security skill by 40 points whenever the key is in your possession

It is only a skill boost. Doors which require keys will still need their respective key, and you still get an insta-unlock by having the respective key in your inventory.