You need to have silverlight to see skydrive UI, but that is for Windows only. Luckily there is an open-source project which implements most of the silverlight. It is moonlight.
Download this from here and install as per instruction on the same page. It is a Firefox plugin, so I think, you must use Firefox for this. (but may work on chromium also, not tested though).
After installing moonlight plugin, you need to install user-agent-switcher extension in Firefox. Then change the user agent to "Internet Explorer" 7 or 8 (I tested with IE7).
Install the extension by clicking this. Then from Firefox menu → Tools → Default user agent → Internet Explorer → Internet Explorer 7. Then refresh the page to see skydrive UI in your Ubuntu browser!