Salam alaikom
First, I want to comment on your statement to provide the statements specifically from the Qur'aan. We as Muslims have to except the Qur'aan and the Sunnah. They are inseparable and both cited when making rulings. For example, no one would say teach me to pray only with references from the qur'aan as that is not possible. We need qur'aan and sunnah just like we need food and water.
“And tell the believing women to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts) and not to show off their adornment except only that which is apparent (like both eyes for necessity to see the way, or outer palms of hands or one eye or dress like veil, gloves, headcover, apron), and to draw their veils all over Juyoobihinna and not to reveal their adornment except to their husbands, or their fathers, or their husband’s fathers, or their sons, or their husband’s sons, or their brothers or their brother’s sons, or their sister’s sons, or their (Muslim) women (i.e. their sisters in Islam), or the (female) slaves whom their right hands possess, or old male servants who lack vigour, or small children who have no sense of feminine sex. And let them not stamp their feet so as to reveal what they hide of their adornment. And all of you beg Allaah to forgive you all, O believers, that you may be successful”
[al-Noor 24:31]
The root of the discussion is what the meaning of the word "Juyoobihinna" is. Below are some ahadeeth on the matter:
It was narrated from Safiyyah bint Shaybah that ‘Aa’ishah (may Allaah be pleased with her) used to say: When these words were revealed – “and to draw their veils all over Juyoobihinna (i.e. their bodies, faces, necks and bosoms)” – they took their izaars (a kind of garment) and tore them from the edges and covered their faces with them.
Narrated by al-Bukhaari, 4481
It was narrated from ‘Urwah that ‘Aa’ishah said: The Messenger of Allaah (peace and blessings of Allaah be upon him) used to pray Fajr and the believing women would attend (the prayer) with him, wrapped in their aprons, then they would go back to their houses and no one would recognize them.
Narrated by al-Bukhaari, 365; Muslim, 645.
It was narrated that Asma’ bint Abi Bakr said: We used to cover our faces in front of men.
Narrated by Ibn Khuzaymah, 4/203; al-Haakim, 1/624. He classed it as saheeh and al-Dhahabi agreed with him.