Prophets can see angels is their natural form.
Al-Bukhari recorded that 'Aishah narrated that the Prophet (SAW) saw angel Jibreel twice in his natural form.
However, ordinary people (i.e., non Prophets) cannot see them unless they (angels) take on a visible form. Angels can indeed take a material form, such as that of a handsome man. An example of this is found in the Quran where Allah sent angels to give Prophet Ibrahim (Abraham) (AW) glad tidings of a son, and then to assist the prophet Lut (LOT) (AW). Lut people, being homosexuals, lusted after these angels, but Allah destroyed the depraved sinners.
Allah says: salam (i.e., greetings of peace) He answered! "salam" See Sura Hud, 11:69-83
Also in the Qu'ran, there is an the incident in which angel Jibreel came to Maryam (Mary) the mother of Isa (Jesus) (AW) to inform her that she would miraculously give birth to 'Isa (Jesus)
Allah says: "Then we sent to her our spirit (angel Jibreel) and He appeared before her in the form of a man in all respects. (Sura Maryam, 19:17-21)
Lastly, there is the famous "Hadith Jibreel" related by Muslim, in which angel Jibreel came to the Prophet (SAW)in the form of a man, and asked him about Islam, Iman, and Ihsan.
This was done in public view, and its purpose was the teach the people about their religion.