Matrices belonging to restricted Lorentz group $SO^+(1,3)$

My Professor says that all members of the restricted Lorentz group $SO^+(1,3)$ may be written as $e^Gamma$, where
$$ Gamma^{mu}{}_{nu}=Lambda^{mu rho} eta_{rho nu}.$$
Here $Lambda$ is an antisymmetric Matrix, and $eta$ is the standard Minkowski metric [$diag(1,-1,-1,-1)$].
I want to prove a weaker statement. That all matrices of the said form belong to $SO^+(1,3)$.
I have been able to show that they belong to $SO(1,3)$. However, the orthochronous bit is troubling me.
Any help is appreciated.

Progress so far:
$$(Lambda eta)^T=-eta Lambda$$
$$implies (e^{Lambda eta})^T=e^{-eta Lambda}$$
$$eta e^{-eta Lambda} eta=e^{-Lambda eta}$$
$$eta e^{-eta Lambda} eta e^{Lambda eta}=I$$
$$e^{-eta Lambda} eta e^{Lambda eta}=eta$$
$$implies (e^{Lambda eta})^Teta e^{Lambda eta}=eta.$$

Hence $e^{Gamma} in O(1,3)$.
Further, Suppose $Lambda$ is written as
$$
begin{bmatrix}
0&vec{lambda}
-vec{lambda}&R
end{bmatrix}
$$
Where $R$ itself is antisymmetric.
Block Multiplication on the right by $eta$, gives $Lambda eta$ to be,
$$
begin{bmatrix}
0&-vec{lambda}
-vec{lambda}&-R
end{bmatrix}.
$$
Clearly Trace of $Lambda eta$ is 0. Hence $Det(Lambda eta)=1$.
Therefore $e^{Gamma} in SO(1,3)$.