The Talmud comes to the conclusion that this commandment prohibits us from asking a salesperson about an item that we don't intend to buy, and also a warning against telling a convert about the misdeeds of his or her ancestors (B. Bava Metzia 58b). It is a sort of catch-all commandment that prohibits any way that we can use words to cause hurt to another person. "Do not wrong one another" can mean almost anything.
Maybe that's the point. We should live our lives in constant concern for the wellbeing of others and with consideration of the many different ways our words could be hurtful. That's a hard way to live, but I have known people who have gone to extraordinary lengths to do so.
Several years ago, an elderly friend of mine was in a nursing home where she was visited regularly by an aide who would trim her toenails and bandage sores on her feet. My friend was concerned because she saw that the aide was not doing the job well. She worried: should she tell the aide not to cut her nails and risk hurting the woman's feelings (and possibly her own health)? Should she inform the nursing home staff about her problems with this aide and risk that the aide might lose her job? Eventually, she chose to talk with the aide and explained the difficulty to her in a kind and encouraging way, to help her to do her job better.
I think that this is exactly what the Torah has in mind when it says, "Do not wrong one another." It is asking us to go to great lengths to keep ourselves from harming other people, to take the time to weigh the consequences of our words, and to tremble a bit at the responsibility we have to others. When we do that, we will be behaving in way that reflects the awe of God.