Why is torque $r \times F$ instead of the reverse? As we all know $\tau = r \times F$.
But why wouldn't we write $\tau = F \times r$?
As it's vector product, these two expressions are actually opposite of each other.
Mathematically
$$r \times F = -F \times r$$
Why is it $\tau = r \times F$ and not $F × r$?
 A: The formula for torque is ultimately set by an arbitrary choice in the way that we define angular velocity.
We have chosen to define angular velocity according to the right-hand rule - in other words, we have arbitrarily chosen that counterclockwise motion corresponds to an angular velocity vector pointing upward. Defining angular velocity this way means that it must be proportional to $\mathbf{r}\times\mathbf{v}$ and not the other way around. Note that the definition of the cross product is also dependent on the right-hand rule, hence is also determined by an arbitrary choice. So these two choices give four possible cases, each of which yields identical physics:


*

*Angular velocity is right-handed, cross product is right-handed $\implies$ $\vec{\omega}\propto\mathbf{r}\times\mathbf{v}$

*Angular velocity is left-handed, cross product is right-handed $\implies$ $\vec{\omega}\propto\mathbf{v}\times\mathbf{r}$

*Angular velocity is right-handed, cross product is left-handed $\implies$ $\vec{\omega}\propto\mathbf{v}\times\mathbf{r}$

*Angular velocity is left-handed, cross product is left-handed $\implies$ $\vec{\omega}\propto\mathbf{r}\times\mathbf{v}$
Aesthetically, there's likely a desire to keep the conventions of angular velocity and the cross product consistent with each other, but there's no reason why they couldn't be chosen to be different.
Anyway, setting the handedness of angular velocity also determines the handedness of angular acceleration $\vec{\alpha}=\frac{d\vec{\omega}}{dt}$ and all higher derivatives, along with angular momentum $\mathbf{L}=I\vec{\omega}$. This, in turn, sets the handedness of torque $\vec{\tau}=\frac{d\mathbf{L}}{dt}$.
A: Angular momentum is given as $\textbf{L}=\textbf{r}\times\textbf{p}$. Thus, we obtain that torque, defined as $\frac{d\textbf{L}}{dt}$ is $\textbf{r}\times\frac{d\textbf{p}}{dt}=\textbf{r}\times\textbf{F}$. As you know, the $\frac{d\textbf{r}}{dt}\times \textbf{p}$ factor vanishes because it is the cross product of two parallel vectors. 
Now, the reason as to why angular momentum is defined as the way it is defined is because it is a physically significant quantity, namely, that it is conserved. But, one can say that we can define the angular momentum with the opposite sign and it would still be a conserved quantity. And then we can define torque also with a negative sign while maintaining that torque is $\frac{d\textbf{L}}{dt}$. 
So, the question comes down to why not define the angular momentum with the opposite sign? Well, it is defined the way it is defined because of the right-hand screw rule which says that when the fingers of the right hand are curled in accordance with the circular motion of a particle, the direction of the thumb ought to align with the direction associated with the angular momentum of the particle. So, the convention is ultimately coming from the right-hand thumb rule. 

Mathematically, this can be traced back to the convention used for the Levi-Civita symbol. In particular, in generic dimensions, angular momentum is defined as $L_{ij}=x_ip_j-x_jp_i$ so there is little sign convention because $L_{ij}=-L_{ji}$ so you have the same physical information with both the plus and the minus sign in the angular momentum matrix anyway. So by choosing the different convention, you'd just be flipping the matrix about diagonal (for no good reason). What happens in three dimensions is that we map the angular momentum matrix to an angular momentum (pseudo)vector because the number of independent components of the angular momentum matrix is $3$ in $3$ dimensions. It is $\frac{n(n-1)}{2}$ in general for $n$ dimensions. So, how do we do this mapping in $3$ dimensions? By defining $L_{ij}=\epsilon_{ijk}L_k$. So, if you want to live in a world where cyclic permutations correspond to a minus sign, you can choose an opposite sign convention for the Levi-Civita symbol and obtain an opposite sign convention for angular momenta and torque. Of course, you can also sneak in a minus sign in the definition $L_{ij}=\epsilon_{ijk}L_k$ itself and define $L_{ij}=-\epsilon_{ijk}L_k$ and then you can get your reversed sign convention without changing the sign convention for the Levi-Civita symbol. 
