Outside, watch the scene between Zelda, her guardian, and you. Some scalding words are exchanged between the latter two before they depart. You can find some fairies in the back of the room, if you want to know.
Anyhow, finish up your business, then go to the platform opposite the crest and perform a Skyward Strike. Soon thereafter, you’ll receive the Amber Tablet and information on (surprise!) the Lanayru Desert. You’ll be warped outside soon.
Outside, use nearby Bird Statue to save and return to the sky.