Find an orthogonal base of a bilinear form on a field of characteristic 2

Question

Let $K$ be a Field of characteristic $2$. On $V=K^2$ the symmetric bilinearform $\beta (x,y) = x_1y_2+x_2y_1 $ is defined.

Now i have to either find an orthogonal base of $V$ or show that such a base doesn't exist. Any tipps or ideas on how to do this?

Thanks in advance.

Answer 1

Hint: Notice that the bilinear form is a determinant; there are no linearly independent vectors $(x_1,x_2)$ and $(y_1,y_2)$ for which $x_1y_2 - y_1 x_2 = 0$