Vectorize else-if statement function using numpy

I have an array of 3 dimensional vectors vec and I want to find a perpendicular vector res_vec to each of those vectors respectively. Using other methods I got some nummerically unstable behaviour so I just check for the smallest component of that vector and set it to zero, while exchanging the two components that are left and negating one of them. However, this is not the main concern, it seems to work just right but it is slow... So my question is, if my code/functionality can be rewritten so I can eliminate the for-loop and vectorize it using some clever numpy-tricks. So far I failed at all attempts doing so. But I'm sure some of you guys can show me some clever ideas.

This is the code:

for i in range(10000):
    index_min = np.argsort(np.abs(vec[i]))
    if index_min[0] == 0:  # x smallest magnitude
        res_vec = np.array([0, -vec[i][2], vec[i][1]])
    elif index_min[0] == 1:  # y smallest magnitude
        res_vec = np.array([vec[i][2], 0, -vec[i][0]])
    elif index_min[0] == 2:  # z smallest magnitude
        res_vec = np.array([-vec[i][1], vec[i][0], 0])

Thank you in advance and happy new year!