What kind of "color charge" does the adjoint fermion carry?

Let us consider the SU(N) gauge theory. The gauge field is in the adjoint representation (rep).

Well-Konwn: If the fermion is in SU(N) fundamental rep, we know the fermions will form a color multiplet of N-component. For $N=3$, we say that there are 3 colors (r,g,b) of a given fermion $\psi$
$$
(\psi_r, \psi_g, \psi_b)
$$

However, if the fermion is in SU(N) adjoint rep, we know the fermions will form a color multiplet of (N$^2-1)$-component. For N=3, we say that there have 8-component of color multiplet. So,

What kind of "color charge" does each component of adjoint fermion carry? There should be 8 different choices.
$$
(\psi_1, \psi_2, \dots, \psi_{8})
$$
What does the 1,2,3, $\dots$, 8 stand for in terms of color indices?

Are the color charges of adjoint fermions organized the same way as the gluons (which are also in adjoint) as in https://en.wikipedia.org/wiki/Gluon#Color_charge_and_superposition? Does both the adjoint fermions carry a **color and an anti-color** just as a gluon does? Why is that?

How can we read this information of color charges from the adjoint Rep of $SU(3)$ Lie algebra?

p.s. we may say the 8 gluons carry 8 distinct color anti-color pairs:
$$
(r\bar{b}+b\bar{r})/\sqrt{2}, -i(r\bar{b}-b\bar{r})/\sqrt{2}, (r\bar{g}+g\bar{r})/\sqrt{2}
$$
$$
-i(r\bar{g}-g\bar{r})/\sqrt{2},(b\bar{g}+g\bar{b})/\sqrt{2},-i(b\bar{g}-g\bar{b})/\sqrt{2}
$$
$$
(r\bar{r}-b\bar{b})/\sqrt{2},(r\bar{r}+b\bar{b}-2g\bar{g})/\sqrt{6}
$$
**And how about the 8-multiplet of adjoint fermions? What color charge do they carry?**

This post imported from StackExchange Physics at 2020-11-05 13:28 (UTC), posted by SE-user annie marie heart