Why did Aaron make the golden calf right after God had spoken the ten commandments to the Israelites?
When confronted by Moses regarding his sin, Aaron blamed the people into pressuring him. "You know the people, that they are set on evil" (Exodus 32:22). He blamed Moses for taking too long to get back from talking with God. "For they said to me, 'Make us gods that shall go before us; as for this Moses, the man who brought us out of the land of Egypt, we do not know what has become of him'" (Exodus 32:23). But the most foolish excuse that he came up with is that it just happened. "And I said to them, 'Whoever has any gold, let them break it off.' So they gave it to me, and I cast it into the fire, and this calf came out" (Exodus 32:24).
The real problem is that Aaron wasn't willing to make a firm stand against people who wanted to sin. "Now when Moses saw that the people were unrestrained (for Aaron had not restrained them, to their shame among their enemies)" (Exodus 32:25). Aaron wasn't a good leader. I suspect that he thought he could compromise. While he knew it was wrong to create an idol for other gods, I suspect he thought he could get away with making a representation of God for the people because after making the idol Aaron had said, 'This is your god, O Israel, that brought you out of the land of Egypt!' So when Aaron saw it, he built an altar before it. And Aaron made a proclamation and said, 'Tomorrow is a feast to the LORD'" (Exodus 32:4-5). Aaron wasn't saying that the golden calf was a replacement for the true God, he was claiming that this calf represented the true God who had brought them out of Egypt. Further, he declared a feast day to the Lord. That is significant because Aaron said the focus of the feast was the Lord, using God's name, Yahweh.
Isn't that the way most corruption in religion begins? "God didn't really say not to do this, so I'm sure this one change won't make a difference to God." Yet it reveals a lack of full respect for God. Men's desires become more important than what God has said (Galatians 1:6-10).
It never stops with one little change. Note how quickly things got out of hand. The "feast to the Lord" became an orgy. "Then they rose early on the next day, offered burnt offerings, and brought peace offerings; and the people sat down to eat and drink, and rose up to play" (Exodus 32:6). Moses heard singing and saw the people dancing before the calf (Exodus 32:18-19).
This ought to serve as a warning to people not to put their desires above what God has said, but sadly the warning is ignored.