A little lower than the angels

Yet you have made him a little lower than the heavenly beings    and crowned him with glory and honor (Ps 8:5). 

3 But I want you to understand that the head of every man is Christ, the head of a wife[or woman] is her husband, and the head of Christ is God. 4 Every man who prays or prophesies with his head covered dishonors his head, 5 but every wife [or woman] who prays or prophesies with her head uncovered dishonors her head, since it is the same as if her head were shaven. 6 For if a wife will not cover her head, then she should cut her hair short. But since it is disgraceful for a wife to cut off her hair or shave her head, let her cover her head. 7 For a man ought not to cover his head, since he is the image and glory of God, but woman is the glory of man. 8 For man was not made from woman, but woman from man. 9 Neither was man created for woman, but woman for man. 10 That is why a wife ought to have a symbol of authority on her head, because of the angels (1 Cor 11:3-10).

The elliptical reference to angels in v10 is a famous crux. What does Paul mean by that? I think Ps 8:5 is probably the key:

i) Paul uses Ps 8 to gloss Gen 1-2. The glory motif in 1 Cor 11:7 comes from Ps 8:5. And I expect the honor/dishonor motif in 1 Cor 11:4-7 also trades on Ps 8:5. 

ii) Regarding Ps 8:5:

This is a paraphrase of the Genesis creation account: "Let us make human beings in our image." Most scholars recognize in the Genesis statement that the first person plural pronouns, "us" and "our," refer to the heavenly court…the human lacks only a little from the exalted status of the beings in the heavenly court. B. Waltke & J. Houston, The Psalms in Christian Worship: A Historical Commentary (Eerdmans 2010), 267.

Although the text doesn't use the word "angel," it refers to angels–given the allusion to the heavenly court, as well as OT angelology. 

iii) Paul is discussing a creational hierarchy, which dovetails with the creational hierarchy in Ps 8: God the Father, Christ as the Second Adam, angels, men/husbands, women/wives. A descending headship. Each has a distinct role in the economy of creation.  

The reference to the angels is way of positioning women in that creational hierarchy, where angels supply a frame of reference via Ps 8:5. Men and woman need to know their place in the grand scheme of things. They have a regal status, as God's viceregents on earth. But that comes with corresponding responsibilities. And that's a derivative status. They are under authority. They exercise delegated authority. They must defer to the God who conferred on them that lofty station.