Emotional intelligence, like other traits such as general intelligence, extraversion, and openness, can be viewed as one of the many parts of personality (for a contemporary overview of personality psychology, see this 2005 article from the American Psychologist).
Psychologists have developed a variety of methods for assessing individual parts of personality. Generally speaking, a particular method is often developed because of its strengths for measuring a specific class of personality parts; that same method may be less good at measuring other parts of personality. For a journal article discussing the many kinds of data collected by psychological tests, see the article here.
Psychologists tend to employ a particular kind of test when measuring intelligence. The test approach goes by several different names: ability testing, performance testing, or criterion-report testing. Regardless of which name is used, tests of that sort employ items that ask a person to solve a problem, and then evaluate the given response according to its correctness in relation to a criterion. For example, on an intelligence test, a test item might ask what is 70 plus 70. The answer is evaluated according to the correct criterion: an answer of 140.
Such criterion-report testing (ability testing), can be contrasted with self-judgment scales. With self-judgment scales, the individual is asked about his or her own self perception, without any check as to its correctness. Self-judgment scales are of value for measuring internal experiences such as moods and emotions. In fact, a good argument can be made that people feel whatever they say they feel, assuming they are being honest. In such a case, self-judgments may be the most accurate representation of the feeling that is available. An example of a self judgment scale of mood can be found on this site here.
Self-judgment scales are of far less value, however, for assessing intelligence, because people tend to be unaware of the intellectual level at which they are functioning. In fact, the statistical relationship between a person's measured intelligence and their self-judged intelligence is very low, with correlations less than r = .30. For a further discussion of this, see this article on emotional intelligence, or this one on psychological measurement.
Several sections of the Controversies area of this web site include discussions of why ability testing is desirable for testing emotional intelligence, and how self-judgment also can play a supportive role in measuring other variables such as mood and emotion that are related to, but different from, emotional intelligence. For the discussions, click here.
To take the concept of emotional intelligence seriously, therefore, involves constructing a criterion-report scale (ability scale; performance scale) that can measure people's capacity to reason accurately with and about emotions.