Rabbi, I have understood that wearing a kippah is a tradition and not a rule, and it was not worn at the time of the Temple, when only priests would wear them.
The Talmud states that a number of Sages used to cover their heads, to show respect for the Divine Presence (Shechinah) above them (Kiddushin 31). Elsewhere, the Talmud writes that convering one’s head protects a person from sin, describing how a person who by nature would have been a thief, was protected from his fate by virtue of his covering his head as a child (Shabbos 156).
These sources suggest that there is no formal obligation to wear a kippah. However, the matter is subject to a dispute among authorities, some of whom write that there is no obligation, whereas others write that the custom has become obligatory (according to one opinion, this is because it is the way of non-Jews to go with uncovered heads, and therefore prohibited for Jews). Note that even authorities who write that there is no obligation to wear a kippah, write that one must do so when a Shul (see Magen Avraham, Orach Chaim 91:3), or when reciting blessings (Kol Bo 11; or Zarua 2:43; Tur, Orach Chaim 8; Taz 8:3).
It is true that at the time of the Temple officiating priests would wear a head-covering, but I have not seen this mentioned as proofs that other Jews did not.
Sources: Concerning the obligation, or otherwise, to wear a kippa, see Kol Bo 11; Tashbatz 549; Maharshal 72; Darkei Moshe 2:3; Magen Avraham 91:3; Biur Ha-Gra 8:6; Bach 2; Birkei Yosef 2; Chasam Sofer, Choshen Mishpat 191.