Does the relation $\nabla\times\vec{A}\approx\nabla\vec{A}-\left(\nabla\vec{A}\right)^T$ have a proper name?

Question

I discovered that curl seems to have an analog which could be used in dimensions other than n=3. $$\nabla\times\vec{A}\approx\left[\begin{matrix}0&\left(\frac{\partial A_x}{\partial y}\right)-\left(\frac{\partial A_y}{\partial x}\right)&\left(\frac{\partial A_x}{\partial z}\right)-\left(\frac{\partial A_z}{\partial x}\right)\\\left(\frac{\partial A_y}{\partial x}\right)-\left(\frac{\partial A_x}{\partial y}\right)&0&\left(\frac{\partial A_y}{\partial z}\right)-\left(\frac{\partial A_z}{\partial y}\right)\\\left(\frac{\partial A_z}{\partial x}\right)-\left(\frac{\partial A_x}{\partial z}\right)&\left(\frac{\partial A_z}{\partial y}\right)-\left(\frac{\partial A_y}{\partial z}\right)&0\\\end{matrix}\right]=\left[\begin{matrix}\left(\frac{\partial A_x}{\partial x}\right)&\left(\frac{\partial A_x}{\partial y}\right)&\left(\frac{\partial A_x}{\partial z}\right)\\\left(\frac{\partial A_y}{\partial x}\right)&\left(\frac{\partial A_y}{\partial y}\right)&\left(\frac{\partial A_y}{\partial z}\right)\\\left(\frac{\partial A_z}{\partial x}\right)&\left(\frac{\partial A_z}{\partial y}\right)&\left(\frac{\partial A_z}{\partial z}\right)\\\end{matrix}\right]-\left[\begin{matrix}\left(\frac{\partial A_x}{\partial x}\right)&\left(\frac{\partial A_y}{\partial x}\right)&\left(\frac{\partial A_z}{\partial x}\right)\\\left(\frac{\partial A_x}{\partial y}\right)&\left(\frac{\partial A_y}{\partial y}\right)&\left(\frac{\partial A_z}{\partial y}\right)\\\left(\frac{\partial A_x}{\partial z}\right)&\left(\frac{\partial A_y}{\partial z}\right)&\left(\frac{\partial A_z}{\partial z}\right)\\\end{matrix}\right]=\nabla\vec{A}-\left(\nabla\vec{A}\right)^T $$ I’m curious about this relation and its uses. I’m also wondering if I need to flip the sign of the relation. Typically, curl is thought of as $$\nabla\times\vec{A}=\left[\begin{matrix}\left(\frac{\partial A_z}{\partial y}\right)-\left(\frac{\partial A_y}{\partial z}\right)\\\left(\frac{\partial A_x}{\partial z}\right)-\left(\frac{\partial A_z}{\partial x}\right)\\\left(\frac{\partial A_y}{\partial x}\right)-\left(\frac{\partial A_x}{\partial y}\right)\\\end{matrix}\right]$$

Answer 1

The name is the infinitesimal rotation matrix and the proper formula is $$\widetilde{\mathrm{\Omega}}=\frac{\nabla\vec{u}-\left(\nabla\vec{u}\right)^T}{2}$$