Skype on Linux was always a bit of a drag to install, and for some reason a 64 bits version is not available for Fedora. It is unlikely to change now that Microsoft bought the whole shop. Luckily, the old 32 bits version still works, with a few tricks.
After downloading the RPM, install with
yum install skype-220.127.116.11-fedora.i586.rpm
Get the 32 bits versions of these libraries:
yum install libv4l.i686 pulseaudio-libs.i686 alsa-plugins-pulseaudio.i686
And finally, make sure you have these binaries:
yum install ld-linux.so.2 libasound.so.2 libXv.so.1 libXss.so.1 libQtDBus.so.4 libQtGui.so.4