I've done a bit of research on this in the last couple of months and it seems that most everyone is in agreement that the "מו" form is used in poetic contexts to denote the third person. It may very well be a form of archaism only found in Biblical poetry. This is why it is found only in places like Tehillim or Iyov, and in the Torah only within poetic portions like "Az Yashir", "Haazinu", or "V'zos Habracha".
Although it most often comes to denote the third person plural, it can be found occasionally being used as the singular, as in Tehillim 11:7. (See Radak's Michlol.) [I've also seen it claimed (by the author of the Anchor Bible's Psalms) to be occasionally used as the first person plural, but I have found no Jewish sources that agree.]
I have been wondering what someone like Malbim would say about this, given his general approach. However, I have been unable to find anywhere where the Malbim discusses this outright. (I would be grateful if anyone can find if/where he does.)
I have found, though, that R' Yaakov Tzvi Meklenburg, whose purpose in writing his commentary Hak'sav V'hakabbala was similar to that of Malbim, does discuss this phenomenon briefly (along with another common phenomenon we find in poetic contexts of "י"s being added to second person objects or possessives). He writes (Shemos 15:7) that the "מו" form is actually a contraction of the plural (with "מ") and singular (with "ו") third persons. It is then used when we want to imply that many people or things are separate entities in reality but that there is an element of unity (אחדות) among them. He compares this to using the singular pronoun to refer to a plural subject (as in the famous "ויחן שם ישראל") and the קרי וכתיב we often find where a plural noun (like "ידיו") is written as if it were singular ("ידו").