No, or at least not easily. They’d rather you just obfuscated your account details. If you actually want your handle removed, you must contact customer services. From Skype’s FAQ:
Remove any personal details you provided. You can replace your details in the fields with random characters such as “qwerty” or “xxx” because empty fields cannot be saved.
You sure, guys? Rather than simply deleting a few database records, you’d rather that we fill them with garbage? I feel sorry for whoever’s tasked with the next annoyingly global cross-service user account merge.
I suppose this could be Microsoft nobly sacrificing its database’s integrity to thwart its transformation into another privacy-invasion-for-cash scheme, but since changing my account’s name to Marmington Fortespube and signing in after a cookie-wipe, Skype welcomed me by my real name. Perhaps they’re simultaneously running what-you-think-we-have and what-we-actually-have databases after all.