Please give us more information about your problem:
Did you tried to save from Word or uploading an existent doc via browser or via the app? Is the app working at all? Sometimes Skydrive has some problems if a Windows-Phone-device and a surface are connected through the same network (and share the same isp-ip) at one time.
Maybe a screenshot would also help (hold win + volume rocker down)
I tried to replicate this last night (my timezone) and Word RT did save the file without any issues. However, I've been using SkyDrive for a while now,on different devices, i.e., iPhone, HTC Mozart (WP7), HTC One V (ICS), and of course Windows 8 and now the Surface RT.
On saving the file, Word RT will establish the connection two times, that shows up as Contacting Server for Information. Then, it will display the Save As dialog box, with the destination clearly showing a SkyDrive folder (My Documents, in my case). Upon entering the file name and selecting save, it goes into saving the file with a progress animation at the lower right of Word RT.