Is emotional IQ related to social skills in one way or another?

No. Social skill is learned behavior. Emotional IQ is inborn, ability to perceive. The saying "truth has to be realized" truth = social version: realization = emotional piece.
Emotional IQ. There is no research-based concept called emotional iq, and the term comes from a book by psychologist daniel goleman. Nevertheless, your emotional functioning can be a determinate of how you behave and affect your interactions with others.