This is a Medrash that obviously requires some kind of an explanation. First of all, was the aveirah [sin] of the Eigel HaZahav [Golden Calf] based on the fact that they used the word “Eleh“? The aveirah was the fact that they made an Eigel HaZahav altogether, a pseudo avodah zarah. The expression “I was angered with the word eleh” seems a bit imprecise, to say the least. Also, what kind of atonement is it that they used the same word when building the Mishkan? Is this some type of gezeirah shavah? eleh-eleh?
The truth of the matter is that the Peirush Marzu on the bottom of the Medrash interprets this Medrash: The word eleh has a certain connotation. It has the connotation of “I am proud of this!” How do we know that? The Marzu writes that another Medrash says: Rav Yitzchak said, at the time of Creation, it is written “Eleh toldos hashamayim v’ha’aretz (These are the products of the heaven and the earth) when they were created on the day that Hashem, G-d made earth and heaven.” (Bereishis 2:4). The Creator praises them. He says, as it were, “Look! Look at My world. Look at what I have created.” The Creator praises the work of His Hands. This is the connotation of using the word eleh – Look at my accomplishment!
Now we can begin to understand the above-cited Medrash. Of course, it is horrible that they built an Eigel HaZahav. But it added insult to injury when they proclaimed “Eleh Elokecha Yisroel asher he’e’lucha m’Eretz Mitzrayim” as if they were proud of what they did. The Gemara (Berachos 19a) says “If someone sees a talmid chocham commit an aveirah at night, don’t suspect him the next day – for he has certainly already done teshuvah for his aveirah of the previous night.”
We all slip up. If someone sees a talmid chochom who did an aveirah at night, he clearly feels bad about it. Without a doubt, by the next morning, he has already done teshuvah. A talmid chochom may have been momentarily grasped by his Yetzer HoRah, but he immediately makes amends and goes on with his life. On the other hand, the pasuk says “(Why do You stand aloof) when the wicked man glories in his personal desire and the robber praises himself that he has blasphemed Hashem?” (Tehillim 10:3). The rasha is proud of what he did. He regrets it not, but rather he brags about his wicked deeds.
This is the difference. Of course, the Eigel HaZahav was wrong, but dancing around the Eigel and proudly saying “Eleh Elohecha Yisroel” really angered the Ribbono Shel Olam.
They were desperate. They thought they lost their leader and they did not know who would lead them. Fine. They made an Eigel HaZahav. Very bad. But at least there would have been some extenuating circumstances to find some justification for their succumbing to their sense of desperation. But when they danced around it and pointed to it (eleh) with glee – that showed that there was no regret whatsoever. That is what angered the Ribbono Shel Olam.
Eleh pekudei haMishkan shows that they were proud of having built the Mishkan. The money contributed, and the effort put into all the work of construction was something about which they felt very good, and they wanted to brag about it. That was the atonement for the aveirah of the Eigel. (R’ Frand)
