I don't have that text, but your question is standard stuff. I suspect what you are missing is the fast interplay of *D*-dimensional vectors and the *D×D* matrices acting on them, sometimes regarded as vector elements themselves, of a larger vector!

The dimension *D* of the representation is the dimensionality of the "vectors" on which the *D×D* representation matrices act. But *all kinds of objects* "furnish" *D*-dimensional vectors on which the suitable representation matrices act to transform them. "Furnish" means provide components of a large *D*-dimensional vector specifying a possibly new representation.  

Antisymmetric tensors $T^{ij}$ in $SO(N)$ with indices $j=1,...,N$  comprise a set of  $\frac{1}{2}N(N-1)$ independent antisymmetric matrices, so they furnish vectors of an irreducible  *D*-dimensional representation, with $D=\frac{1}{2}N(N-1)$: they define the relevant vector space. Each such matrix is a component of the *D*-vector.

It is also true that  there are $\frac{1}{2}N(N-1)$ generators of $SO(N)$,  indexed by the *(mn)* pairs,
$$\mathcal{J}_{(m n)}^{i j}=\left(\delta^{m i} \delta^{n j}-\delta^{m j} \delta^{n i}\right)$$
which act on the space of *N*-dimensional *j*s to rotate them to such *i*s. 

* However, these *N×N* antisymmetric matrices may also be regarded as the $D=\frac{1}{2}N(N-1)$ components of a *new vector*, acted upon by some larger $\frac{1}{2}N(N-1) \times \frac{1}{2}N(N-1)$ matrices, (actually $\frac{1}{2}N(N-1)$ independent such, indexed by *(mn)*. This is the adjoint representation.) 

So the *D=N* generators became the $D=\frac{1}{2}N(N-1)$-vector elements of the adjoint (which they furnished).  Fortuitously, these two dimensions are the *same* for *N=3*, so the fundamental of SO(3)  happens to also be its adjoint!
 
Now 
> We can also regard the antisymmetric tensor $T^{ij}$ as an $N$-by-$N$ matrix, and hence write it as a linear combination of the $\mathcal{J}_a$'s, with coefficients denoted by $A_a$: $$T^{ij} = A_a \mathcal{J}^{ij}_a.$$


This object is an *N×N* antisymmetric matrix, but ***also*** an $\frac{1}{2}N(N-1)$-dimensional vector in the adjoint representation vector space, introduced above, itching to be rotated by the suitable 
$\frac{1}{2}N(N-1)\times\frac{1}{2}N(N-1)$ adjoint representation matrices. (These matrices are gotten/furnished  by the structure constants of the Lie algebra you'd get from your $\cal J$s in the *D=N* irrep., but don't worry about them.)

* The dimension of the adjoint representation is also the dimension of the corresponding Lie algebra (regarded as a vector space as you saw).