Lie algebra of nonabelian group is $[T^a,T^b]=if^{abc}T^c$. For $SO(3)$ case, is the representation $T^a_{ij}=-i\epsilon^{aij}$ fundamental or adjoint? The fundamental representation is defined as identifying $T^a$ the original generator. The adjoint representation is defined as identifying $T^a$ as structure constant. For $SO(3)$ case, generators seem to be same as structure contant.
Reference:
Defining a Lie algebra by the commutation relations $[T^a,T^b]=if^{abc} T^c$, the adjoint representation is defined by $(T_{adj}^a)^{bc}= if^{abc}$. Now it turns, that in the special case of $so(3)=su(2)$, you have $f^{abc} = \epsilon^{abc}$, where $\epsilon^{abc}$ is the totally antisymmetric tensor. So, your representation is the adjoint representation.
[EDIT]
Due to OP comments, some precisions :
There is a subtelty. the Lie algebra defined by the commutation relations $[T^a,T^b]=i\epsilon^{abc} T^c$ is fundamentally $su(2)$, but it is also $so(3)$.
Why this? In fact, different Lie groups may correspond to the same Lie algebra. In the over way, if we take the "exponential" of a Lie algebra, we obtain only one simply connected Lie group. Starting from the Lie algebra $su(2)$, and exponentiate it, we obtain the simply connected Lie group $SU(2)$. However, from the Lie group $SU(2)$, we may consider other Lie groups which are the quotient of $SU(2)$ by a finite group, so these new groups are not simply connected, but they share with $SU(2)$ the same Lie algebra. For instance, one has $SO(3) = SU(2)/Z_2$. So two elements of $SU(2)$ correspond to the same element of $SO(3)$. But $SO(3)$ and $SU(2)$ share the same Lie algebra $su(2)$
We see that $SU(2)$ play a central role in all this, so it is natural to look at its fundamental representation, which is $2-$dimensional, the fundamental representation is simply given by the Pauli Matrices $(T_{f}^a)= \frac{1}{2}\sigma_a$. The adjoint representation for $SU(2)$ is 3-dimensional and is given by $(T_{adj}^a)^{bc}= if^{abc}$
Now, you may also think at the representations of $SO(3)$, and, yes, the fundamental representation is $3$-dimensional, and is the same as the adjoint representation of $SU(2)$.
It is in fact better to give to $SU(2)$ a central role, in the meaning, that the representations of $SO(3)$ are included in the representations of $SU(2)$ (the inverse is not true), which is more fundamental (because, as explained above, it is the simply connected group you obtain by exponentiating the Lie algebra).
Finally, a problem of vocabulary, maybe you have heard of will hear also about "spinorial" representations of $S0(3)$, but it is in fact simply the representations of $SO(3)$ which are not standard ("vectorial") representations of $SO(3)$, so these are representations of $SU(2)$ which are not standard representations of $SO(3)$. For instance, you may consider the fundamental representation of $SU(2)$ (which applies to a 2-spinor), as a "spinorial" representation of $SO(3)$
