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n=5; p=2
set.seed(713)
X = matrix(rnorm(n*p), n, p); X
           [,1]        [,2]
[1,] -1.5925966  0.37590371
[2,] -0.1458793  0.28096447
[3,] -1.0048927 -0.60433386
[4,] -1.5337787 -0.03059243
[5,]  0.3318419  0.46158609
set.seed(713)
X = matrix(ncol = p, nrow = n)
for (i in 1:p) {
 X[, i] = rnorm(n) }
X
           [,1]        [,2]
[1,] -1.5925966  0.37590371
[2,] -0.1458793  0.28096447
[3,] -1.0048927 -0.60433386
[4,] -1.5337787 -0.03059243
[5,]  0.3318419  0.46158609

n=5; p=2
set.seed(713)            # set seed
X = matrix(rnorm(n*p), n, p); X
           [,1]        [,2]
[1,] -1.5925966  0.37590371
[2,] -0.1458793  0.28096447
[3,] -1.0048927 -0.60433386
[4,] -1.5337787 -0.03059243
[5,]  0.3318419  0.46158609
set.seed(713)           # set same seed again
X = matrix(ncol = p, nrow = n)
for (i in 1:p) {
 X[, i] = rnorm(n) }
X
           [,1]        [,2]
[1,] -1.5925966  0.37590371
[2,] -0.1458793  0.28096447
[3,] -1.0048927 -0.60433386
[4,] -1.5337787 -0.03059243
[5,]  0.3318419  0.46158609

If you replace randomly generated normal variates by sequences of integers, the rules for filling a matrix might be clearer to you:

I'm not quite sure what your difficulty is. If you set the same seed before each program, then you should fill an $n\times p$ matrix by columns with exactly the same normal variates.

n=5; p=2
set.seed(713)
X = matrix(rnorm(n*p), n, p); X
           [,1]        [,2]
[1,] -1.5925966  0.37590371
[2,] -0.1458793  0.28096447
[3,] -1.0048927 -0.60433386
[4,] -1.5337787 -0.03059243
[5,]  0.3318419  0.46158609
set.seed(713)
X = matrix(ncol = p, nrow = n)
for (i in 1:p) {
 X[, i] = rnorm(n) }
X
           [,1]        [,2]
[1,] -1.5925966  0.37590371
[2,] -0.1458793  0.28096447
[3,] -1.0048927 -0.60433386
[4,] -1.5337787 -0.03059243
[5,]  0.3318419  0.46158609

But if I've missed the point and you're asking something else, then I suggest you replace randomly generated normal variates by sequences of integers. Then the rules for filling matrices might be clearer to you:

n = 5; p = 2; matrix(1:(n*p),n,p)
     [,1] [,2]
[1,]    1    6
[2,]    2    7
[3,]    3    8
[4,]    4    9
[5,]    5   10

n = 5; p = 2
X = matrix(ncol = p, nrow = n)
for (i in 1:p) {
 X[, i] = 1:n
}
X
     [,1] [,2]
[1,]    1    1
[2,]    2    2
[3,]    3    3
[4,]    4    4
[5,]    5    5

When matrix turns a vector into a matrix, the default order is to fill the matrix by columns. (All vectors are considered column vectors, even if they print out as rows.)

w = c(1,2,3,4,3,2)
X = as.matrix(w)
X
     [,1]
[1,]    1
[2,]    2
[3,]    3
[4,]    4
[5,]    3
[6,]    2

t(X)  # transpose is a row vector
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    2    3    4    3    2

t(X) %*% X   # matrix multiplication
     [,1]
[1,]   43
X %*% t(X) 
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    2    3    4    3    2
[2,]    2    4    6    8    6    4
[3,]    3    6    9   12    9    6
[4,]    4    8   12   16   12    8
[5,]    3    6    9   12    9    6
[6,]    2    4    6    8    6    4
X*t(X)
Error in X * t(X) : non-conformable arrays
t(w)
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    2    3    4    3    2
t(w)%*%w
     [,1]
[1,]   43

n = 5; p = 2; matrix(1:(n*p),n,p)  # n rows, p col
     [,1] [,2]
[1,]    1    6
[2,]    2    7
[3,]    3    8
[4,]    4    9
[5,]    5   10

n = 5; p = 2
X = matrix(ncol = p, nrow = n)
for (i in 1:p) {
 X[, i] = 1:n
}
X
     [,1] [,2]
[1,]    1    1
[2,]    2    2
[3,]    3    3
[4,]    4    4
[5,]    5    5

When matrix turns a vector into a matrix, the default order is to fill the matrix by columns. (All vectors, unless specifically modified, are considered column vectors, even if they print out as rows.)

w = c(1,2,3,4,3,2)   # 'c' for column vector
X = as.matrix(w)
X
     [,1]
[1,]    1
[2,]    2
[3,]    3
[4,]    4
[5,]    3
[6,]    2

t(X)  # transpose is a row vector
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    2    3    4    3    2

t(X) %*% X   # matrix multiplication
     [,1]
[1,]   43
X %*% t(X) 
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    2    3    4    3    2
[2,]    2    4    6    8    6    4
[3,]    3    6    9   12    9    6
[4,]    4    8   12   16   12    8
[5,]    3    6    9   12    9    6
[6,]    2    4    6    8    6    4
X*t(X)
Error in X * t(X) : non-conformable arrays
t(w)
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    2    3    4    3    2
t(w)%*%w
     [,1]
[1,]   43