Without knowing what its good for and “best” Julia style, this would be a 1:1 conversion:
i_hat1 = [0 1]
j_hat1 = [-1 0]
transform1 = [i_hat1;j_hat1]'
i_hat2 = [1 0]
j_hat2 = [1 1]
transform2 = [i_hat2;j_hat2]'
combined = transform2 * transform1
v = [1, 2]
combined * v