Shell command for the normal program termination of Skype from the command line
As the comments have pointed out, there exist tools to match processes by name. This is roughly equivalent (better than) what you're trying to do:

pkill skype


It's better because:


It's shorter and cleaner
It won't kill itself


By the second point I mean that ps -e | grep skype will match the grep process itself too. That won't happen when using pkill.

Code review

If we wanted to imitate pkill, I would suggest writing like this:

ps -e | awk '/[s]kype/ { print $1 }' | xargs kill


That is:


Replace two processes with one (grep + cut with awk)
Use [s]kype instead of skype is a common trick to make the awk itself not match
Drop -s term from kill, as TERM is the default signal anyway