What is a three dimensional irrep ${\bf 3}$ of $SO(3)$? 
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*What is three dimensional irreducible representation of $SO(3)$ denoted by ${\bf 3}$? Are they vectors or antisymmetric tensors of rank two each of them has three independent components. 

*Also when one writes $${\bf 3}\times{\bf 3}={\bf 1}+{\bf 3}+{\bf 5},$$ do they mean that there is no difference between the transformations of ${\bf 3}$ on the left and the ${\bf 3}$ on the right? In that case, is it true that an antisymmetric tensor transforming as $A^\prime_{ij}=R_{ik}R_{jl}A_{kl}$ can be reduced to $V^\prime_i=R_{ij}V_j$?
 A: Yes, they are the same: every real  antisymmetric tensor of rank 2 in $\mathbb{R}^3$ is one-to-one related with a real  vector with $3$ components and this association commutes with the corresponding action of $SO(3)$.
A: What is ${\bf 3}$?
One example is:
$$ Y_1^{\pm 1}(\mathbf {r}) \propto (x \pm iy)/r $$
$$ Y_1^0(\mathbf {r}) \propto z/r $$
which transform with the Wigner D-Matrix:
$${\displaystyle Y_{1 }^{m}({\mathbf {r} }')=\sum _{m'=-1 }^{1 }[D_{mm'}^{(1)}({\mathcal {R}})]^{*}Y_{1 }^{m'}({\mathbf {r} }),}$$
If you look at the 9 dyads:
$$ Y_{1 }^{m}({\mathbf {r} })Y_{1 }^{m'}({\mathbf {r} })$$
you can find 5 linear combinations, such as:
$$  Y_1^1(\mathbf {r}) Y_1^1(\mathbf {r}) \propto (x+iy)^2 \propto Y_2^2(\mathbf {r})$$
which transform as the $\bf 5$ dimensional representation. Another combination looks like $x^2+y^2+z^2$, which is scalar, and the remaining 3 transform like $\bf 3$, but look a little different, for example:
$$ Y_1^1(\mathbf {r}) Y_1^{-1}(\mathbf {r})- Y_1^{-1}(\mathbf {r}) Y_1^1(\mathbf {r})\propto x^2 + y^2 + i(-yx+xy) - [x^2 + y^2 +i(yx - xy)]$$
$$ = i(xy-yx)$$
transforms like $Y_1^0$.
