Your formula
$$\frac{1}{2}\nabla v^2-v\cdot \nabla v = v\times \nabla \times v$$
is a special case of the following,
as copied literally from a Wikipedia page :
$$
\nabla(\vec{u}\cdot\vec{v})=
(\vec{u}\cdot\nabla)\vec{v}+(\vec{v}\cdot\nabla)\vec{u}+
\vec{u}\times(\nabla\times\vec{v})+\vec{v}\times(\nabla\times\vec{u})
$$
Step by step for your formula:
$$
\frac{1}{2} \nabla (\vec{v} \cdot \vec{v} )= \frac{1}{2} \left[ \begin{array}{c}
\frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{array} \right]
\left(v_x^2 + v_y^2 + v_z^2 \right) = \left[ \begin{array}{c}
v_x \frac{\partial v_x}{\partial x} +
v_y \frac{\partial v_y}{\partial x} +
v_z \frac{\partial v_z}{\partial x} \\
v_x \frac{\partial v_x}{\partial y} +
v_y \frac{\partial v_y}{\partial y} +
v_z \frac{\partial v_z}{\partial y} \\
v_x \frac{\partial v_x}{\partial z} +
v_y \frac{\partial v_y}{\partial z} +
v_z \frac{\partial v_z}{\partial z} \end{array} \right]
$$
$$
(\vec{v}\cdot\nabla)\vec{v} =
\left(v_x \frac{\partial}{\partial x} + v_y \frac{\partial}{\partial y} + v_z \frac{\partial}{\partial z} \right)
\left[ \begin{array}{c} v_x \\ v_y \\ v_z \end{array} \right] =
\left[ \begin{array}{c}
v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} + v_z \frac{\partial v_x}{\partial z} \\
v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y} + v_z \frac{\partial v_y}{\partial z} \\
v_x \frac{\partial v_z}{\partial x} + v_y \frac{\partial v_z}{\partial y} + v_z \frac{\partial v_z}{\partial z}
\end{array} \right]
$$
Left hand side:
$$
\frac{1}{2} \nabla (\vec{v} \cdot \vec{v} ) -
(\vec{v}\cdot\nabla)\vec{v} =
\left[ \begin{array}{c}
v_y \frac{\partial v_y}{\partial x} + v_z \frac{\partial v_z}{\partial x}
- v_y \frac{\partial v_x}{\partial y} - v_z \frac{\partial v_x}{\partial z} \\
v_x \frac{\partial v_x}{\partial y} + v_z \frac{\partial v_z}{\partial y}
- v_x \frac{\partial v_y}{\partial x} - v_z \frac{\partial v_y}{\partial z} \\
v_x \frac{\partial v_x}{\partial z} + v_y \frac{\partial v_y}{\partial z}
- v_x \frac{\partial v_z}{\partial x} - v_y \frac{\partial v_z}{\partial y}
\end{array} \right]
$$
Definition of cross product:
$$
\vec{a}\times\vec{b} = \left[ \begin{array}{c}
a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x
\end{array} \right]
$$
Right hand side:
$$
\vec{v}\times(\nabla\times\vec{v}) = \vec{v} \times \left[ \begin{array}{c}
\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \\
\frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \\
\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}
\end{array} \right] =
\left[ \begin{array}{c}
v_y \frac{\partial v_y}{\partial x} - v_y \frac{\partial v_x}{\partial y}
- v_z \frac{\partial v_x}{\partial z} + v_z \frac{\partial v_z}{\partial x} \\
v_z \frac{\partial v_z}{\partial y} - v_z \frac{\partial v_y}{\partial z}
- v_x \frac{\partial v_y}{\partial x} + v_x \frac{\partial v_x}{\partial y} \\
v_x \frac{\partial v_x}{\partial z} - v_x \frac{\partial v_z}{\partial x}
- v_y \frac{\partial v_z}{\partial y} + v_y \frac{\partial v_y}{\partial z}
\end{array} \right]
$$
You can check out (any typos?) that left hand side and right hand side are equal.
Its' a tedious job though, even more if you try to do it for the Wikipedia formula.