I have a doubt about the relation between the structure constants of a Lie group and the representation of its algebra. Let's see an example:
For $SU(3)$ the number of generators is 8 and we write them as $\{T_a, a = 1, ..., 8\}$. Then, the adjoint (matrix) representation of these elements is
$$T_a \rightarrow [t_a^{Adj}]_{ij} = f_{aji}, \quad [T_a, T_b] = f_{abc}T_c \quad (1)$$
(We suppose implicit the sum over $c$)
From Eq. (1) and the fact of having 8 generators (so $i, j$ stand for 1 to 8), each $t_a^{Adj}$ is a 8x8 matrix, but we are used to the Gell-Mann 3x3, not 8x8, matrices as the generators' representation.
Therefore, what is happening here?
Thanks in advance!