Bi-vector in Minkowski space

I have a problem, I have a bi-vector that define like:

$\omega^{\mu \nu}=a^{\mu}b^{\nu}-a^{\nu}b^{\mu}$

where,

$a^{\mu}=(a^0,a^1,a^2,a^3)$ and $b^{\nu}=(b^0,b^1,b^2,b^3)$

I need show that $\Lambda^{\mu}{}_{\nu}=\exp(\alpha \omega^{\mu}{}_{\nu})$ and the explicit form like a matrix.

$$\Lambda^{\mu}{}_{\nu}~=~\exp(\alpha \omega^{\mu}{}_{\nu})\qquad\qquad (\leftarrow \text{wrong})$$
$$\Lambda^{\mu}{}_{\nu}~=~\exp(\alpha \omega)^{\mu}{}_{\nu}~=~\delta^{\mu}_{\nu}+\alpha \omega^{\mu}{}_{\nu}+ \frac{\alpha^2}{2}\omega^{\mu}{}_{\lambda}\omega^{\lambda}{}_{\nu}+{\cal O}(\alpha^3).$$