Cross product of multi-variate Gaussians and their expectations

Let $$a, b \in \mathbb{R}^3$$ be two vectors, chosen independently from multi-variate Gaussian distributions ($$a \sim N(\mu_a, \Sigma_a), b \sim N(\mu_b, \Sigma_b)$$).

I'm trying to find a closed-form expression for $$\mathbb{E}[(a \times b)(a \times b)^T]$$ (expectation of outer product of cross products).

While I already found that $$\mathbb{E}[a a^T] = \mu_a \mu_a^T + \Sigma_a$$, a neat closed-form for $$\mathbb{E}[a \times b]$$ still eludes me.

I've tried to find relevant identities and properties but so far found almost nothing on the combination of cross products and distributions.

EDIT: I suspect $$\mathbb{E}[a \times b] = \mathbb{E}[[a]_\times b] = \mathbb{E}[[a]_\times] \mathbb{E}[b]={[\mu_a]}_\times \mu_b = \mu_a\times\mu_b$$ (where $$[\cdot]_\times$$ is the cross product matrix).

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