The general equation is the same for both the univeriate and multivariate cases,

$$ V[\hat{\beta}] = V[(X^{T}X)^{-1}X^{T}Y]\\ = (X^{T}X)^{-1}X^{T}V[Y]X(X^{T}X)^{-1}\\ = \sigma^2(X^{T}X)^{-1} $$

This is unfortunately challenging to calculate. Examining $X^{T}X$ we see (assuming an intercept is included in the model),

$$X^{T}X = \begin{bmatrix} n & \sum_{i=1}^{n}x_{i1} & \sum_{i=1}^{n}x_{i2} & \ldots& \sum_{i=1}^{n}x_{ik}\\ \sum_{i=1}^{n} x_{i1} & \sum_{i=1}^{n} x_{i1}^2 & \sum_{i=1}^{n} x_{i1}x_{i2} & \ldots & \sum_{i=1}^{n}x_{i1}x_{ik} \\ \vdots & \vdots & \vdots & & \vdots\\ \sum_{i=1}^{n}x_{ik} & \sum_{i=1}^{n} x_{ik}x_{i1} & \sum_{i=1}^{n} x_{ik}x_{i2} & \ldots & \sum_{i=1}^{n} x_{ik}^2 \end{bmatrix}$$

which is symmetric, but does not have a clean expression for the inverse. Therefore the simple formulas for the univariate case are not available for the multivariate case.

The general equation is the same for both the univariate and multivariate cases,

$$ V[\hat{\beta}] = V[(X^{T}X)^{-1}X^{T}Y]\\ = (X^{T}X)^{-1}X^{T}V[Y]X(X^{T}X)^{-1}\\ = \sigma^2(X^{T}X)^{-1} $$

This is unfortunately challenging to calculate. Examining $X^{T}X$ we see (assuming an intercept is included in the model),

$$X^{T}X = \begin{bmatrix} n & \sum_{i=1}^{n}x_{i1} & \sum_{i=1}^{n}x_{i2} & \ldots& \sum_{i=1}^{n}x_{ik}\\ \sum_{i=1}^{n} x_{i1} & \sum_{i=1}^{n} x_{i1}^2 & \sum_{i=1}^{n} x_{i1}x_{i2} & \ldots & \sum_{i=1}^{n}x_{i1}x_{ik} \\ \vdots & \vdots & \vdots & & \vdots\\ \sum_{i=1}^{n}x_{ik} & \sum_{i=1}^{n} x_{ik}x_{i1} & \sum_{i=1}^{n} x_{ik}x_{i2} & \ldots & \sum_{i=1}^{n} x_{ik}^2 \end{bmatrix}$$

which is symmetric, but does not have a clean expression for the inverse. Therefore the simple formulas for the univariate case are not available for the multivariate case.