On sesquilinear forms

Question

I was trying to solve some exercises related to sesquilinear forms:

Let $V$ be a $\mathbb C$-vector space.

1. Prove that the set $\mathcal{S}(V)$ of sesquilinear forms on $V$ is a vector subspace of the $\mathbb C$-vector space of all maps $\psi: V \times V \to \mathbb C$.

2. Prove that the set $\mathcal{H}(V)$ of Hermitian sesquilinear forms on $V$ is a vector subspace of the $\mathbb R$-vector space $\mathcal{S}(V)$. Is $\mathcal{H}(V)$ a $\mathbb C$-vector space of $\mathcal{S}(V)$?

3. Prove that we have a direct-sum descomposition of $\mathbb R$-vector spaces:

$$\mathcal{S}(V)=\mathcal{H}(V) \bigoplus i \mathcal{H}(V)$$

I didn't have any problem with the first one, but with the second one I'm a bit confused because of course $\mathcal{H}(V)$ is a vector subspace of $\mathcal{S}(V)$, but I don't see why should or shouldn't be a $\mathbb C$-vector space. And for the third one is only necessary to show that every sesquilinear form can be see it as the sum of two Hermitian forms?

I would really appreciate any advice or hint you could give me. Thanks.