Error in EKF documentation about ahrs HOT 1 OPEN

Indeed, a skew symmetric matrix looks like:

$$ \left[ \tilde{a} \right] = \left[ \begin{array}{c} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{array} \right] $$

Just for clarity:

$$ 2 \dot{q}= \left[ \begin{array}{c} q_0 & -q_1 & -q_2 & -q_3 \\ q_1 & q_0 & -q_3 & q_2 \\ q_2 & q_3 & q_0 & -q_1 \\ q_3 & -q_2 & q_1 & q_0 \end{array} \right] \left[ \begin{array}{c} 0 \\ \omega \end{array} \right] = \left[ \begin{array}{c} -q_1 \omega_0 - q_2 \omega_1 - q_3 \omega_2 \\ q_0 \omega_0 - q_3 \omega_1 + q_2 \omega_2 \\ q_3 \omega_0 + q_0 \omega_1 - q_1 \omega_2 \\ -q_2 \omega_0 + q_1 \omega_1 + q_0 \omega_2 \end{array} \right] = \left[ \begin{array}{c} 0 - \omega_0 - \omega_1 - \omega_2 \\ \omega_0 + 0 + \omega_2 - \omega_1 \\ \omega_1 - \omega_2 + 0 + \omega_0 \\ \omega_2 + \omega_1 - \omega_0 + 0 \end{array} \right] q = \left[ \begin{array}{c} 0 & -\omega^T \\ \omega & \left[ \tilde{\omega} \right] ^T \end{array} \right]q $$

$$ -\left[ \tilde{\omega} \right] = \left[ \tilde{\omega} \right]^T $$

from ahrs.