There is no instruction in the Quran for wearing a hijab. The Quran tells us to be modest and draw a veil over our breasts. In other words, to be modest in our clothing, not to walk around with out bodies exposed.
The hijab was imposed by man due to his weaknesses and oppression. I do not wear hijab and have no intention of doing so. Hijab is a cultural innovation of man. If it was obligatory to cover our heads and faces, why is it not prescribed in the Quran??

And tell the believing women to lower their gaze and to be mindful of their chastity, and not to display their charms [in public] beyond what may [decently] be apparent thereof; [37] hence, let them draw their head-coverings over their bosoms. [38]

In the interpretation it says:
The noun khimar (of which khumur is the plural) denotes the head-covering customarily used by Arabian women before and after the advent of Islam. According to most of the classical commentators, it was worn in pre-Islamic times more or less as all ornament and was let down loosely over the wearer’s back; and since, in accordance with the fashion prevalent at the time, the upper part of a woman’s tunic had a wide opening in the front, her breasts cleavage were left bare. Hence, the injunction to cover the bosom by means of a khimar, (a term so familiar to the contemporaries of the Prophet) does not necessarily relate to the use of a khimar as such but is, rather, meant to make it clear that a woman’s breasts are not included in the concept of “what may decently be apparent” of her body and should not, therefore, be displayed

So the bottom line is that according to Quran in normal public situations the woman is to cover her breasts and there is nothing about covering the head while during prayer it just says to “beautify yourself” and nothing about covering the head:

7:31 (Asad) O CHILDREN of Adam! Beautify yourselves [23] for every act of worship, and eat and drink [freely], but do not waste: verily, He does not love the wasteful!

someone wrote a comment about how you wouldn’t wear a track suit to a job interview and that you’d wear something formal like a nice clean suit (kind of like beautifying yourself). But using the same analogy, wearing a hijab with the track suit doesn’t make it a formal/appropriate attire for the interview and therefore seems to be irrelevant. I don’t see how that’s being used as an argument to cover the head during prayer because if its just about being formally/nicely dressed than you can do that without covering your head too.