$$\nabla \times V= \hat{e_x}\space(\frac{\partial}{\partial{y}} V_z-\frac{\partial}{\partial{z}} V_y)+\hat{e_y}\space(\frac{\partial}{\partial{z}} V_x-\frac{\partial}{\partial{x}} V_z)+\hat{e_z}\space(\frac{\partial}{\partial{x}} V_y-\frac{\partial}{\partial{y}} V_x)$$
In the above provided equation, I have understood "$\nabla \times V$" to be just a notation (which is defined to have above value) and not to indicate any cross product.
Reference: Is $\nabla$ a vector?
The following has been extracted from the book "Mathematical methods for Physicists" by Arfken, Weber, and Harris:
Another possible operation with the vector operator r is to take its cross product with a vector. Using the established formula for the cross product, and being careful to write the derivatives to the left of the vector on which they are to act, we obtain $$\nabla \times V= \hat{e_x}\space(\frac{\partial}{\partial{y}} V_z-\frac{\partial}{\partial{z}} V_y)+\hat{e_y}\space(\frac{\partial}{\partial{z}} V_x-\frac{\partial}{\partial{x}} V_z)+\hat{e_z}\space(\frac{\partial}{\partial{x}} V_y-\frac{\partial}{\partial{y}} V_x)$$
$$= \begin{array}{|ccc|} \hat{e_x} & \hat{e_y} & \hat{e_z} \\ \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}} \\ V_x & V_y & V_z \end{array} \\\\\space Eq (3.58)$$
This vector operation is called the curl of V. Note that when the determinant in Eq. (3.58) is evaluated, it must be expanded in a way that causes the derivatives in the second row to be applied to the functions in the third row (and not to anything in the top row); we will encounter this situation repeatedly, and will identify the evaluation as being from the top down.
I have trouble in understanding the above passage. I feel the determinant equivalent of the curl of V to be wrong, because we don't get the curl of V when we solve that determinant. Is it right to write in determinant form?
The determinant expression is notation that encodes a mnemonic, just like $\nabla \times$ does. Both notations are abuses in the sense applying the usual definition of $\times$ (a product of vectors) or $\det$ (a scalar determined by a matrix) here is nonsense---$\nabla$ isn't a vector, and $\hat{e}_x$ and $\frac{\partial}{\partial x}$ (etc.) are operators, not numbers. In both cases, the notation reminds us of the (slightly complicated) form of the curl.
In the case of the determinant mnemonic, we simply expand across the first row, applying the operators in the second row to the scalar (functions) in the third row, and multiply the results by the appropriate vectors in the first row; NB that the latter is multiplication of a scalar by a vector, not of two scalars, so this too is an abuse despite that we call both operations multiplication.