It is appropriate for this kind of Hebrew genealogy to skip generations, and Matthew outright admits he wants to make this three sets of 14 generations.
Timescale and comparison to Luke has me convinced Matthew is also skipping generations from the Captivity to Jesus. And I think even the Hebrew Bible genealogies skip some generations from Nasshon to Boaz.
All that said why leave these three of all of them out is something worth enquiring about.
The usual explanation for Matthew 1:8 skipping right from Joram to Uzziah is that the three kings skipped (Ahaziah, Joash and Amaziah) were “bad” kings. But they weren’t the worst kings, in fact Joash and Amaziah could be considered pretty okay. Manasseh was way worse but Chronicles records his repentance, however Amon and Ahaz were also worse than those three and have no repentance recorded. I’ve seen one suggestion that it has to do with them starting good but then “going bad” and not repenting, but that narrative describes Solomon more than anyone else, and can arguably apply to Uzziah and Josiah as well.
I’m intrigued by the fact that all three are consecutive, it’s like an entire era of the Davidic Monarchy’s history is being skipped.
What’s also interesting is that the two names flanking this skip are both names for which the Hebrew Bible has a degree of chaos concerning what their name even is. Uzziah is also called Azariah and Joram is also called Jehoram. And in each case at least one of those names is shared by other individuals who lived at the same time they did.
I’ve been considering the possibility that the post Pentateuch Historical books of the Hebrew Bible should perhaps not be considered as authoritative or infallible as those books much more directly quoted as Scripture by Jesus or other New Testament writers. They are not unambiguously quoted in the NT, New Testament references to the “Old Testament” are focused on the Law and the Prophets. When Paul talks about “rightly dividing the word of truth” and about testing all things, maybe it’s okay to subject the historical books to the same historical criticism other historical texts are given.
Immanuel Velikovsky in the first volume of Ages in Chaos in the section on the Amarna letters deals with what he perceives to be contradictions in the text by suggesting that Jehoram King of Israel didn’t exist. I however, even back when I wasn’t open to what I’m suggesting now figured a far more plausible theory would be to say it's the Jehoram of Judah who didn’t exist. That perhaps both Kings and Chronicles in their final forms are Judean histories seeking to erase that there was a time when Judah was conquered by the House of Omri.
That perhaps 2 Kings 8:16 was originally saying the fifth year of Jerhoram of Israel’s reign in Israel was when he became King of Judah. And 2 Kings 8:25 that in his twelfth year he made his son Ahaziah King of Judah similar to how the heir to the throne of England is the Prince of Wales. And Jehoram was perhaps married to his sister which was acceptable in some ancient Near Eastern pagan monarchies.
And so when Jehosheba and her husband the Priest Jehoiada conspire against Athaliah to put a kid on the Throne, it’s their kid not a Nephew. The Azariah who lived at the same time as King Uzziah was a “chief priest” in 2 Chronicles 14-20, a term that is distinct in Hebrew from “High Priest” but many treat it as the same.
Maybe Uzziah was the first patrilineal descendant of Jehoshaphat to sit on the Throne of David since Jehoshaphat was alive and that is what Matthew 1:8 is actually telling us?