No. There was literally just a box to type a name in and a “rename” button. Not any type of “file save” dialog box which lets you select a drive and path, etc.
This seems to be a 8.1 “feature.” See the following, for example:
http://techjia.com/where-are-files-stored-for-windows-sound-recorder-windows-8-1/
Or this “helpful” Microsoft help file:
http://windows.microsoft.com/en-us/windows-8/sound-recorder-app-faq
“Your recordings are stored within the Sound Recorder app. If you uninstall the app, your recordings will be deleted. To send the recording file to someone through another app, use the Share charm.”