In Fanon's Text Black Skin, White Masks, Fanon describes the self as something that is influenced by outside sources, or the other. More specifically, Fanon believes that the colonial systems of race in the Antilles have led black people to strive towards being seen as white—literally defining their individual success through how well they can assimilate with the other. For example, in chapter 2, "The Woman of Color and the White Man," Fanon explicitly defines the self in relation to the other as a
motion toward the world and toward his like. A movement of aggression, which leads to enslavement or to conquest; a movement of love, a gift of self, the ultimate stage of what by common accord is called ethical orientation. Every consciousness seems to have the capacity to demonstrate these two components, simultaenously or alternatively. The person I love will strengthen me by endorsing my assumption of my manhood, while the need to earn the admiration or the love of others will erect a value-making superstructure on my whole vision of the world... the fact remains that true, authentic love—wishing for others what one postulates for oneself, when that postulation unites the permanent values of human reality—entails the mobilization of psychic drives basically freed of unconscious conflicts.
Left far, far behind, the last sequelae of a titanic struggle carried on against the other have been dissipated.
Fanon writes a couple of chapters detailing this search for validation in interracial relationships, where both black men and women seek out relationships with white women and men as an effort to separate themselves from their blackness. If your particular interest is in the topic of the self and other in Fanon's text, I would encourage you to read chapters 2 and 3 closely. In these chapters, Fanon details the process of how the black self is changed and colonized in relation to whiteness. These changes of the self involve speaking the colonizer's language, changing the manner of dress to match the colonizer, associating with other highly educated black people or white people as a means of being accepted by the colonizer, and incorporating the black self fully into white society, such that even a white man would describe the black man as "white" in spite of his skin.
After undergoing this process of colonization, the black man now has two selves which are conditionally based on whom he is around: the black self, which is shunned and only shown to one's black family (and even then it is toned down), and the white self, who speaks French (or any colonizer's language), marries a white woman, has white children, and is welcomed into white society as long as he does not display the black self. This is where the title, Black Skin, White Masks, becomes especially relevant to the content, as a black person must mask the black components of their self with white traits in order to assimilate with the other.