You have seen

Thus you shall say to the house of Jacob and tell the children of Israel: "You have seen what I did to Egypt, and how I lifted you up on wings of eagles, and brought you to Me." (19:4)

Thus you shall say to the children of Israel: "You have seen that I have spoken with you from heaven." (20:18)

The phrase "You have seen" used to begin a speech is quite rare in Tanach. Besides these two verses, it is used once each by Moshe and Yehoshua - each time to introduce a covenant with God made before their deaths - and once by Yirmiyahu. So its appearance twice in succession is somewhat striking.

The first instance (19:4) refers to the exodus from Egypt, and comes to introduce the revelation at Sinai.

The second instance (20:18) refers to the revelation at Sinai, and comes to introduce the commandments in parshat Mishpatim (and the last few verses of Yitro).

Seeing as two of the other three uses of this phrase introduce the making of covenants, it is not surprising that the same is true here. The revelation at Sinai is called a covenant (Devarim 5:2), and the commandments in Mishpatim are called "the book of the covenant" (Shemot 24:7).

Moreover, we learn that the events in question are not independent of each other, but rather form a linear progression. The exodus from Egypt created a sort of relationship between God and Israel. But it then had to be made explicit by the acceptance of the Ten Commandments. Even that was not enough, and the covenant had to be more fully fleshed out, with a long list of the laws we will keep, and the rewards we will get for it.

That's all I have to say about the similarity between these two phrases. Now, a few words on the difference between them, as interpreted by the midrash.

There is one noticeable difference: the audience. The first phrase is spoken to both "the house of Jacob" and "the children of Israel". The second is only spoken to the latter.

On a pshat level you might say that "the house of Jacob" and "the children of Israel" mean the same thing, and the first verse uses both for some stylistic reason, perhaps to provide extra formality. But the midrash cannot accept such a "fuzzy" answer. In midrash, every word of the verse should add something meaningful and definite.

So therefore, the midrash (quoted by Rashi) says that "the house of Jacob" means the women, and "the children of Israel" means the men. I had always thought this identification was for linguistic reasons: "bnei yisrael" can mean either "children" of Israel or simply "sons" of Israel, while "the house" is a place where women were likely to be found.

But now, we can add a structural reason for this identification. (Alas, you will see, it is no more politically correct than the linguistic one.) It stands to reason that women were equal participants in accepting the Ten Commandments and the basic commitment to Jewish faith. But since they are not commanded to learn Torah to the same extent as men, perhaps the numerous halachic details of parshat Mishpatim are not directly relevant to them. Therefore, the midrash may be thinking, women are mentioned as "the house of Jacob" in the introduction to the Ten Commandments, but not in the introduction to Mishpatim.