Install an old version of Skype, like from here Uninstall the old version of Skype, removing everything (like with the free version of Revo Uninstaller) Download and install the latest version of Skype
Do note that 1603 seems to be a general error code for any Windows installer failure (for example, I got this error code when another instance of msiexec.exe was running), so this may not actually be your issue. Check the error message to be sure you're fighting the right battle.
Speaking of Skype fail, when Skype updated for Windows 7 (which was a while ago), they took away the ability to minimize to the clock and instead always pinned to the taskbar. I despise having something pinned to the taskbar AND being down by the clock (makes no sense to me to have something taking up 2 spots). You can get around this by right clicking on the Skype icon in your start menu, hit properties, then under the compatibility tab run it in compatibility mode for Windows Vista.
Edit (5/26/2016): as of now, they at some point added the "keep skype in the taskbar while I'm signed in" option that fixes my last complaint.