Download the Skype 4 rpm from the skype web site at this link http://www.skype.com/intl/en/get-skype/on-your-computer/linux/downloading.fedora. There is only the 32-bit version available. Therefore, the instructions below will apply to 32-bit or 64-bit Fedora Linux.
To install Skype to your computer, run the following from the command line / shell:
# yum localinstall skype-220.127.116.11-fedora.i586.rpm<enter>
The advantage of using yum to install skpe is that it will automatically detect, download and install all the required dependencies.
Once you are done, you can start Skype on your Fedora Linux desktop and use it. If you face an issue with no sound when you talk during the Test call, you should do:
# yum install alsa-plugins-pulseaudio.i686 alsa-lib.i686<enter>