orthonormal basis

give an orthonormal basis for the $gl(3)$-module $V(\epsilon_1 - \epsilon_3)$. give the matrix of the generators $a^1_2, a^2_1, a^1_1$ and $a^2_2$ in the corresponding representation, and give the branching rule for $gl(2) \subset gl(3)$ in this case.

ok. so I know that $gl(3)$ has basis $\{a^1_1, a^1_2, a^1_3, a^2_1, a^2_2, a^2_3, a^3_1, a^3_2, a^3_3\}$, that $V$ is a $gl(3)$-module if $\forall \alpha, \beta \in \mathbb{C}, v, w \in V, x_1, x_2 \in gl(3)$ linearity and bracket preservation hold, and I know that $\epsilon_1 - \epsilon_3 = (1, 0, -1, 0, 0, 0,...)$

because the basis of $gl(3)$ has 9 elements, does that mean that the orthonormal basis for the $gl(3)$-module $V(\epsilon_1 - \epsilon_3)$ will also have 9 elements? and does $\epsilon_1 - \epsilon_3 = (1, 0, -1, 0, 0, 0, 0, 0, 0)$? im also a little confused as to what the $gl(3)$-module $V(\epsilon_1 - \epsilon_3)$ actually is...any help AT ALL will be greatly appreciated!